Well you do have a strange problem ...
First off there should be no corrolation between running the A/C and Micro, and having the 15A Conv breaker tripping unless your Panel scheule is labled incorrectly. It almost appears that the Micro is on the Conv breaker.
As a test, turn off the Conv breaker and try to operate the Microwave. If it does not operate then the labels are wrong.
If your RV is like mine, the Conv breaker (15amp) has a duel purpose. It not only has to provide power to the Converter but also all Non-GFI outlets in the RV. If you run a 1500W heater (12.5amps) that is plugged into a Non-GFI outlet that only leaves 2.5amps for the Converter before the 15amp breaker will trip. You can confirm this by removing the plastic cover on the Converter and look at the Conv breaker you will probably see 2 wires coming out of it. Whoops: Word of caution first. UN-PLUG RV FROM IT's POWER SOURCE FIRST BEFORE OPENING UP THE PANEL!!!! Work by flashlight.
Notice that all other breakers only have one wire.
Now look and see that one of the 2 wires coming off the Conv breaker will go to a taped up wire nut and then it continues to romex cable that goes to all Non-GFI outlets. If this is the case then it is really easy to fix by installing a dedicated breaker for the Non-GFI outlets. I did this yesterday to my RV.
I picked up a duel 15A breaker from HD for about 15 bucks. Install it in the next location of the breaker panel. It literally just snaps in.....
Now, go back to those 2 wires coming out of the Conv breaker and relocate the one going to the taped wire nut and land it on the new 15A breaker. Walla! dune.
What this does is dedicates the Non-GFI outlets to their own 15A breaker ... same with the Converter... it has its own (original) 15A breaker. No sharing!
BTW: Be careful when adding up "Amps" to compute total usage when including DC (lights, etc) appliances. As an example, you Converter with a 15Amp (AC) breaker has a total of 1800W (15A x 120Volts) capacity. The converter, "coverts" 120V AC to 12v DC to provide the 12V to the RV. Theoretically, 1800W at 12V is 150Amps. Your converter is rated for probably only 55A if it's like mine. What Im getting at is if you add up all of the 12V DC lights amps usage you can easyly surpass 15amps but it won't pop the 15A AC Conv breaker because it's on the output of the converter at 12v.
Anyway, if you dry camp alot like me then "Yes" it's a great idea to replace all bulbs with LEDS. The batteries will llast longer.
Sorry as this post turned out to be MUCH longer than I planned.
1970 F250 Highboy 4x4
2013 Cougar 21RBSWE