Never EVER backfeed.. Please do not Backfeed, The device used to backfeed is called a SUICIDE plug because it KILLS people DO NOT do it.
Now that we have that
It is very likely that your inverter has a single outlet and someone installed an extension cord, if after market.. If not after market (if factory installed) it's not big enough to run more than the Front end electronics and is hard wired in the power line to there.
Properly doing it is going to take a major re-wire
And.. You may overload either the inverter or the breaker that feeds it when on shore power.
So, find out how much inverter you have
If you have enough (At least a kilowatt, two would be better) then the path is
120 volt service---Inverter with built in ATS --- Sub panel
If inverter does not have built in ATS (Automatic transfer switch)
Inverter ---To ATS
LIne from breaker box---To ATS
ATS ---- Sub panel
Sub panel to all loads you wish to inverter power
NOTE> ATS can be a Manual Transfer Switch too.. Just not a el-cheapo double throw from Radio shack, Use a proper transfer switch.