The first I would do is test for GFCI trips. Goto to WALMART and pick up one of their RV30A-15A adapter cables.
Looks like this...

Then connect this to one of the 120VAC receptacles on the inverter to determine if the Inverter you have will live with your trailer GFCI receptacles.
If your trailer GFCI breakers trip then you will have to solve that problem first before trying to wire in your INVERTER into your Trailer Power Distribution Panel.
Wiring in the Inverter into your power distribution panel is not an easy thing to do. This requires removing wiring from the panel and going through an approved SELECTOR SWITCH or AUTO RELAY function to switch your internal wiring to either select the shore power or the Inverter circuits.
The problem you will have is that some items in your trailer draw too many WATTS to operate from your Inverter so you will want to leave those items connected to your Power Distribution Panel.
There are ELECTRICAL CODE LAWS against using suicide "widow Maker" double MALE adapters from a 120VAC source. Any thing that has 120VAC present has to be a socket not a plug. i.e. this would be an illegal wall socket even though you can purchase one...

This would be an interesting project to install a MANUAL SELECTOR switch assy like this photo shows. The GEN would be your INVERTER.

Most folks would probably want all of this to switch automatically but you still have to address what you will want to have powered up from your inverter circuit.
Or you can do a simple install of long extension cord type mulitport 120V receptacle placed perhaps at your home entertainment cent location and maybe the bedroom night stand area and just have the emergency power receptacles available to use when the shore power is off... All cables hide behind cabinets or the use of WIREMODE cable management items from LOWES. This layout is just for a 300WATT to a 600WATT PSW Inverter setup.

You also have to consider what the INVERTER is going to draw from your battery setup. Inverters are very power hungry... a 2500WATT inverter will need to have almost 300AMPS of 12VDC power to operate under full load. That is a lot of power to draw from your batteries. You will need several batteries installed to run this large of an inverter... Installing more capacity batteries also means you have to increase your on-board charging capability with a bigger converter/charger unit.
Nothing is free it seems...
Roy ken