If you have typical battery set up on the tongue of the trailer, mounting the inverter as close as possible in the front compartment is ideal. Check the installation instructions of your inverter and use the recommended cables and fuse to connect to the batteries. I think you'll find that you will be using 1/0 cable and a 120 amp fuse.
Microwave and hair dryer are big loads, it will challenge a 1500 W inverter with typical battery bank. The hair dryer or coffee maker will be OK, as will any entertainment devices.
FWIW, I run a small 120 V refrigerator and a few 120 V entertainment devices, along with all the cabin loads using a 1000W Xantrex PS inverter. I have 300 AH of battery capacity in two banks of batteries. I also have solar charging that helps during the day.
My 12V loads are small, only some LED lighting and occasional water pump and fans. My alarms are all battery powered and I don't have any other standby loads. My "furnace" doesn't use a fan/blower.
I just dedicated a few receptacles for 120av inverter use and my inverter has 5V USB outlets for charging phones.