Can I run the refrig on AC from an inverter in the tow vehicle?

Thanks for all the quick and helpful comments on my 12V battery contention question. Texas Camper hit the nail on the head with his question. Why would I want to do this? Here are my thoughts. First, I saw at least a rumor that the new TM can have shore power connected while closed down. Second, I am a DRY camper and am interested in multiple days without recharging (2*T106 batteries). Third, I have read most of the posts regarding driving with refrigerator on (12V) and arriving with a LESS THAN FULLY CHARGED TM battery (due to voltage (12A refrigerator) drop between the TV alternator and the TM battery I am convinced).

So my next request is for comments on the following:
Starting my trip with a fully charged TM battery.
(a) I pull the fuse in the TV which supplies 12V to the 7 pin trailer connector.
(b) I install a 110VAC inverter in TV (say 500w)
(c) While travelling, I connect output of inverter to TM (as in shore power)
(d) I Run the TM refrigerator on 110VAC (I'm guessing 200 watts)
(e) The TM battery will be charged by the TM converter.

Seems this would allow me to arrive with FULLY charged TM battery. Must be a big flaw in this scheme somewhere. (I know the TM battery must be fully charged to start else the 500W inverter will choke)
Thanks for all your great comments.