I need to replace the #2 fuel injector. Is this something a layman such as I can accomplish?

Possible causes
- Faulty Fuel Injector
- Faulty Fuel Injector Interface Module
- Fuel Injector harness is open or shorted
- Fuel Injector circuit poor electrical connection

Your injectors are shot sounds like.

all the layman can do change fuel filter and bleed air from the system and make sure the delivery pump is working ok.

It just can't do that. No way to shut down one side of an engine.
Misfires are caused by lack of fuel, spark, or compression. One of the three is missing. One whole side-that rules out spark. Coil pack design just won't affect one side of the engine. That leaves fuel or compression. Some multi-port injection systems are so-called "bank fired" -that is, one side or bank of the engine will pulse the injectors at the same time, first one bank, then the other. But in the early '90's, some cars came out with sequential fuel injection, which pulses the injectors individually, following the firing order of the cylinders, more precise fuel metering that way. I would have thought that an Avalon would be sequential injection, but you could check. Anyway, you need to check if the injectors are working on 2, 4, & 6. The power wire to all injector systems is a single wire that is spliced to each injector. Conceivably, the splice to that bank or side may have broken. With key on, every injector should have voltage at the injector connector. Only two wires to each injector- a power wire and a ground wire. If no power on 2, 4, or 6-that is probably it. (On bank fired injection, all three ground wires on each side splice together before the ground wire goes into the computer. With sequential, each injector ground wire goes to the computer separately.)
Beyond fuel, it is possible that the fuel/air mixture is being affected on only one side. If you have a noisy exhaust leak on that side, it could lean out the mixture.
Finally, low compression will cause a misfire. Possibly, a camshaft on that side is not in time. Only way to check is to do an engine compression test.

try some fuel injector cleaner or replace fuel injectors or fuel filter

Sounds like something in fuel system is clogged form running to low on fuel if vehicle is old and fuel filter has not been replaced for a long time.
It does not take much but a small particle to clog injector or it could be fuel pump is going out. You need to check fuel pressure and change filter and also you might have to get fuel system flushed and you might just have replace injector at worst. Hope this helps

Are you sure that the fuel pump is properly installed? Have you tested the fuel pump circuits to see if they are working properly? Did you also replace your fuel filter? (It could be totally clogged up) Have you checked to see if there is any fuel pressure at the fuel rail or are you just assuming that the pump is not delivering fuel to the injectors? Have you checked your fuel injector circuits for power and for injector pulse from the EngineControl Module? Have you scanned your Engine Control module to see if you possibly have a Crank Sensor or Cam Sensor problem?

The oil pressure switch only keeps the fuel pump running once the engine starts. The fuel pump should run for 2-3 seconds when you turn the ignition switch to the "Run" position to prime the injectors. The oil pressure switch is the problem only if the engine will not continue to run on its own after it starts up.

I have Fuel Injected Porsches ... Older & newer than yours...
Both rely heavily on proper FUEL PRESSURE...

... and the factory manuals for both...
HIGHLY RECOMMENDED

I always suggest covering the BASICS first...

Make sure you are getting fuel & fire.

The are simple pressure gauges & spark lights
to accomplish both...

Even if you can hear & feel the injectors clicking...
MEANS absolutely NOTHING without pressure.

1979 PORSCHE 924 2 0L L4 Fuel Pump RockAuto

P0174 System Too Lean Bank 2
P0171 System Too Lean Bank 1

Have you tried to work on the fuel system side? Have you tried
fuel injector cleaner from a bottle into the gas tank?

i.e., fuel filter, fuel pump, fuel pressure readings from the rail,
fuel injectors:
Fuel Injectors Testing The easiest way to test the operation of the fuel injectors is to listen for a clicking sound coming from the injectors while the engine is running. This is accomplished using a mechanic’s stethoscope, or a long screwdriver. Place the end of the stethoscope or the screwdriver (tip end, not handle) onto the body of the injector. Place the ear pieces of the stethoscope in your ears, or if using a screwdriver, place your ear on top of the handle. An audible clicking noise should be heard; this is the solenoid operating. If the injector makes this noise, the injector driver circuit and computer are operating as designed. Continue testing all the injectors this way.
CAUTION
Be extremely careful while working on an operating engine, make sure you have no dangling jewelry, extremely loose clothes, power tool cords or other items that might get caught in a moving part of the engine.
All Injectors Clicking If all the injectors are clicking, but you have determined that the fuel system is the cause of your driveability problem, continue diagnostics. Make sure that you have checked fuel pump pressure as outlined. An easy way to determine a weak or unproductive cylinder is a cylinder drop test. This is accomplished by removing one spark plug wire at a time, and seeing which cylinder causes the least difference in the idle. The one that causes the least change is the weak cylinder.
If the injectors were all clicking and the ignition system is functioning properly, remove the injector of the suspect cylinder and bench test it. This is accomplished by checking for a spray pattern from the injector itself. Install a fuel supply line to the injector (or rail if the injector is left attached to the rail) and momentarily apply 12 volts DC and a ground to the injector itself; a visible fuel spray should appear. If no spray is achieved, replace the injector and check the running condition of the engine.
One or More Injectors are not Clicking Unplug the fuel injector connector
Probe the two terminals of a fuel injector to check its resistance
Plug the correct "noid" light directly into the injector harness connector
If the correct "noid" light flashes while the engine is running, the injector driver circuit inside the PCM is working
If one or more injectors are found to be not operating, testing the injector driver circuit and computer can be accomplished using a "noid" light. First, with the engine not running and the ignition key in the OFF position, remove the connector from the injector you plan to test, then plug the "noid" light tool into the injector connector. Start the engine and the "noid" light should flash, signaling that the injector driver circuit is working. If the "noid" light flashes, but the injector does not click when plugged in, test the injector’s resistance.
If the "noid" light does not flash, the injector driver circuit is faulty. Disconnect the negative battery cable. Unplug the "noid" light from the injector connector and also unplug the PCM. Check the harness between the appropriate pins on the harness side of the PCM connector and the injector connector. Resistance should be less than 5.0 ohms; if not, repair the circuit. If resistance is within specifications, the injector driver inside the PCM is faulty and replacement of the PCM will be necessary.

A bad fuel injector can cause spark plug issues. If the injector does not supply enough fuel, the cylinder(s) without enough fuel will run hot, often damaging the spark plugs.

But your mechanic needs to make sure that the injector is the issue and not a fuel pressure issue from regulator, filter, or pump. Hopefully he/she has already done those checks.

As far as the labor cost being less the spark plugs wouldn't be removed during the fuel injector replacement. Even if they did, most mechanics won't cut the price. But if you are a regular customer it can't hurt to ask.