If all you can access is a 15 amp receptacle for your extension cord, it is what it is and you have to work with it.....
Notwithstanding voltage drop issues, you should be able to run your AC, TV and lights. If you wanted to run a toaster, coffee maker, etc., all you would need to do is shut the AC off while the appliances are on. No big deal.
If I were you, I would take a voltmeter and check the source voltage that you are plugging into. If it is low there, then it will be even lower inside your RV. If low enough, it could damage your AC. Even if the source voltage *seems* okay, sometimes you can have a long branch circuit run from the main panel to the receptacle which will result in voltage drop. Then you would have an extension cord plus your 30 amp shore power cord. You could easily be looking at 150`or more, which can result in too low a voltage at the RV panel.
I would go with a heavy duty 15 amp extension cord that has 12 gauge wire. A 25 foot long one would be a good idea to keep in your trailer. You can use it for plugging something into the exterior recept. on the RV or as an extension cord cord for your 30 amp cord along with an adapter. You could also carry a 25 foot 30 amp extension cord in your RV (assuming the extra weight is okay for you). One day you may find yourself at a campsite and your standard RV cord won`t reach.
You might consider a hard-start capacitor and relay. Lot's of info. available via google like this for example:
AC unit capacitor and relayOne tip I`d like to add is never, ever plug your RV cord into anything unless the source power is turned off. This can be the cg pedestal breaker or the one in the house panel. If you plug in live, you will hear a snap, crackle and pop, and in the dark, see a flash. This is inrush current to the invertor and if you do it often enough, you will burn out a plug and/or receptacle. It causes pitting of the contacts which ends up increasing contact resistance. I have burned out a couple of inexpensive (ie., cheap) extension cords at home by being too lazy to turn off the breaker. I had one laying in the driveway that almost caught fire except that someone noticed it smoking away..... If you can`t avoid doing this, use a 30/15 amp adapter on your RV cord and a 15/15 amp adapter on the extension cord which will be sacrificial. We now have a full 30 amp receptacle in our carport fed by #8 (50A) gauge wire for most of the way to the panel and a disconnect switch in the carport.