battery for inverter

I'm with the others here. Most likely your issue is cables not big enough (thickness) and/or too long.
Inadequate cable is the main issue with getting inverters to deliver power. You need really ridiculously huge ones.
I'd add heavier ones in addition to keeping your existing cable to. More is better here and there's no such thing as too much.
We're talking 00-gauge, REALLY heavy ones.
If you can get your inverter within a couple feet of the batteries (which is a very good idea) you can get away with 2-gauge, if you triple them up.

My 1000W PSW inverter runs my 21 cuft refrigerator plus a DVR without any problems. X 7 on your cables being to small.

One thing to keep in mind with ANY inverter is to properly size the cables from the inverter to the batteries, pick up the positive lead from the farthest positive post and the negative from the opposite farthest negative post, same applies to charging from a converter or solar and be sure the input cables for the inverter have an inline fuseable link with a rating just below the maximum amp draw of the inverter. Running undersize cables can cause a fire if you pull maximum amps from the inverter.

Your batteries should do OK....just to add to the conversation, I have a residential Refer, a portable ice maker, a mini fridge, a wine cooler etc etc that require 120 volt.... I usually run these all off my 2500 watt inverter but what I have noticed is at idle, I'm not getting enough amps out of the alternator.

Generally I wouldn't worry about this but with recent issues with my Allison Tran's not getting enough voltage, I'm in the market for a higher capacity alternator. The MH originally had a 220 amp alternator(Leece Nevell)...after many issues I ended up with a 125/150 amp that has done the job prior to all the junk I have running. Good thing I run the generator alot!!! :-)

OH, getting ready to add a Norcold 3.2 CF Freezer only(great deal at Adventure RV@ $205.00)...another reason for larger capacity alternator...also have 6 6 volt batteries.

Two golf batteries should easy start the fridge.
How long and what size wire connects the 12 volt battery to inverter?

How are the batteries getting charged while driving? Unless they are also the engine batteries I wouldn't count on much charging from the alternator.

Two 6 volt batteries should work just fine (if they are not discharged) for the inverter to run the fridge. What size cable are you running from the batteries to the inverter?

Starting here - "the 2 6volt batteries would be charged while driving"

Possibly. How much is delivered to the batteries? How long you drive on average?