Why do I have to replace the 9 volt battery in my guitar every 2 weeks?

Hi!
At this point, I would just take it to a tech rather than poke around in it. It sounds like the power section is the problem as you have guessed, but without the technical knowledge you'd be shooting in the dark. That particular amp is right on the line of being expensive enough that would be worth repairing, but you might do just as wel to check Ebay for another used one.
Good Luck!

no it won't buy new batteries

This has the earmarks of battery problem... Supposedly this uses a 9 volt battery... You CANNOT use rechargeable batteries in these as the actual voltage of the rechargeable is lower than alkaline and they will fail quickly and as they age you may get only seconds of operation. When they go off the battery will recover a bit and even look like it comes on again. Do NOT count on the battery LED to give actual answer. When you turn on the unit, read the battery voltage under load with a meter. It should be at least 8.5 volts. NEXT possibility: A chronic problem with body packs is the battery contacts. There is a fine line between having good contact and being able to get batteries out. I have seen where people stuff paper in to increase tension on the contacts.

Make sure you use Alkaline battery and NOT rechargeable type... the rechargeable battery will NOT work in the pedals (their voltage is lower even though they are labeled as 9 volt replacement).

Other internal problems can be due to cracked circuit board or broken solder joints. These are very common as people step on the cables when plugged in. This breaks the connection inside. A fuzzy sound often occurs when one of the contacts is marginal due to bad solder at the jack terminals.

Possible that the output tubes that were replaced are drawing too much current, hence the popping sound (may be "motorboating", one common word to describe a similar sound)

This means the tubes need to be biased properly. You need to get it to someone who knows what they are doing, ie a good amp tech, not GC. The voltages here are dangerous.

There is little data on the internet about this... However my AEL20E uses 2 AA cells. Some older ones may have used a 9 volt battery. Yours should use the 2 AA cells which go into the box right by the XLR output plug.

It is important to turn off the battery when not in use... they may have a switch OR often when you plug the guitar cable into the unit that completes the circuit to the battery. Another thing the 9 volt batteries are not the greatest. With continuous pwer I would expect the battery to last a couple days. Remember that the devices that have a pushbutton to alternately turn on the device DO still place a small current drain on the battery... For those devices REMOVE the battery when unit is be idle for a week or more.

Your guitar has active pickups. Active pickups require a 9-volt battery to operate effectively. They are louder and have more "punch" than passive pickups. If your not into swapping 9 volts fairly often then you can swap out you pickups for some passive ones. If your going to be storing your guitar for a while you will want to remove the battery to prevent unnecessary drain and oxidation of the battery.

There is a battery compartment in the EQ, 9 volts some of them have a battery test lite , not all no need to charge just replace it.
This is the best I can do for now with the info you gave. good luck. I have been playing and repairing for 35 yrs. need any thing more just ask. Good luck DFD please rate me TK U

Take it to a guitar shop and have them check out the soldering and connections within the guitar.
Martin has a record of cutting corners on their less expensive models.