LADBS Plumbing inspector says we need a temperature regulator for an In-Wall Lavatory Kit in a tub. The tub does not have a shower nor a wand? Which section of the code indicates this?

Bath sink will be on what is called a "wet vent" for the shower.
That means they are on the same drain.
There should be a cleanout on the sink drain below where the drain enters the wall. That would be your "last resort" access to put a snake in to break up the obstruction.
I would try using a plumbers suction plunger on the shower drain.
Show no mercy!
Cheers.
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I'm a retired plumbing inspector so should be able to coach you.

while the wall is open is the time to replace the plumbing it's cheaper to do it know then in a few weeks when its all closed in $100 now or $500 later

The most common reason for basement leakage is from the seal at the drain or the seal behind the overflow. With the overflow, the water trickles down after bouncing off your body and back onto the tub. This possibility MUST be eliminated first before determining the next step, Turn on the cold water and fill the tub and let the water go into the overflow for about 5 minutes. This will simulate someone in and using the shower. Check for leaks in the basement.

This test, as simple as it is, will also implicate bad plumbing connections at the valve itself behind the wall as well as eliminate backsplash from the showerer bouncing back onto your control valve(s).

If you have a leakage, wait a day and, with the drain open, turn on the hot/cold valve(s). and waste some water for another 5 minute test. This test eliminates the overflow as a suspect, and with the drain open almost always eliminates the drain as well. Hey..in plumbing there are no perfect answers.

If you still have leakage and no one is showering you have pretty well isolated the valve or connections to the valve. If you don't have leakage, it points to water bouncing off the body and onto the control valve(s) at which point resealing would be in order.

You can get leakage also if the caulking around the tub where the tub meets the wall has gone south. Stamped out steel tubs (common in rental apartments) have open corners so if the wall-to tub caulking is poor then the shower bounce off the body will easily trickle to the basement. This is not an issue with cast iron or fiberglass/plastic tubs.

If these tests truely point to the valve(s), repost your ? and refer to the drain tests completed, and supply as much information as you can about the brand and age of your valve(s)

plumbing code varies by area, so I don't think it's an issue with the company. they do have vacuum breakers 4 sale, though.
http://www.signaturehardware.com/product598

tighten the stem if its at its tightest and stem still loose then get a rebuild or repair kit/parts most plumbing and hardware stores carry them

If it is a two handle tub valve, then more likely no. But if you want to be sure, get into shower, a nice hot one and then have someone flush the toilet you will find out real quick.

Lowes might carry them but i know a plumbing supply house does for sure. Take center caps off hot and cold handls, then screws everything else should pull right off.

sounds like there may be some blockage in pipes. try to run a long snake through both drains or run all fixtures in bathroom and try plunging vigorously in tub and shower while running water. if nothing still, it may be possible that your underground plumbing is back graded or incorrect fittings and plumbing installation.

only when its on? it can only be from 3 places the the plumbing up to shower head or the tub spout or crack in fixture make a hole in the wall behind fixture easier then replaceing tile if you can turn on and see where the water is coming from you need to open anyways thank you for useing fixya please rate me

the kit to rebuild the peerless diverter is very common. take the cartridge to any hardware store and ask the plumbing clerk to assist you in getting the correct kit. good luck