Inverters come in two types. (times two types) in this case the two types I will discuss are Stand alone and inline
Stand alone have DC in and AC out,, In line have AC in AC out and DC in/out (same connection works both ways)
If you have an in-line, then when you run the honda, it will switch the inverter to standby and the inverter will pass power to the TVs from the honda.
IF you have a stand alone, then you will either have to run it to a transfer switch of some kind, in which case see transfer switch below OR you will have to run it to separate outlets. (Or locate it where the power cords can reach the outlets on it however I do not recommend this).
IF you use the separate outlet method (make them red or orange) then the TV and Sat receiver will operate off the inverter full time, and the presence of the honda will not matter.
IF you use a transfer switch. Read on
Transfer switches:
these come in two types Automatic and Manual. ONE form of manual is described above (separate outlets)
Another is a proper DPDT cener off break before make Transfer Switch, TRANSFER switches often need to be double tapped to switch from one source to the other this is a safety feature.
This is the kind of switch I use for one circuit in my rig.
The AUTOMATIC switches (I have one of those too well two technically) sense when power is coming in on one of the two lines, (Usually that input is marked GENERATOR) You run shore power to this input, and the inverter to the other input, if shore power (Such as from the HONDA) is present it will switch to it, if not, it Drops back to the inverter
For your install, I recommend the outlet method.