Well.... I'll try to help if I can.
When you say
I do have experience with CNC machines and operating them
do you mean that you have built one yourself?
See, the problem is mainly the working are you mentioned. For a commercially available cutter you are looking in the $1500-2000 range (50 watt CO2 lasers) which is probably what you want, but WAY over your budget. See below for an example of my cheap Chinese machine, US based will be at least twice that figure (but higher quality...)
You could go with a really cheap K40, but the working area is about 1/4 what you are wanting (but the prince is right at around $300.)
Here's an example of someone on the forums that has started on this journey;
https://forum.flitetest.com/index.php?threads/so-i-bought-a-cheap-40w-laser.55664/
I don't think he is expanding the working area though, he has a needle cutter based on the MPCNC for foamboard.
Here's a link the probably the longest running forum on here;
https://forum.flitetest.com/index.php?threads/cutting-foam-sheets-with-a-needle.24251/
Some might suggest diode lasers, but I don't think they will have the power you are really going to want.
So, IF (and I do mean IF) you are comfortable building a larger XY system, you could rip the guts out of the K40 and get the working are you want.
Know that CO2 lasers are INVISIBLE and can seriously harm you, your eyesight, cause fires... the list goes on (so know the risks!) AND be prepared to build a FULL enclosure (out of metal... CO2 lasers won't cut metal.)
That said, there is NOTHING like spending 20 minutes cutting foam and having a full kit ready to go!! (it really is awesome!)
FYI, this is something like what I have. They are REALLY 50 watt machines (per the tube length) but you see them everywhere as 60s...
I just upgraded mine and had to build an extension 'cause the tube sticks out about half a foot with a real 60 watt tube.
https://www.ebay.com/itm/Laser-Engr...=item2cb907ddcc:g:YDEAAOSwdnZaJPiQ:rk:13:pf:0
The K40 will have (30-40 watt) will have more than enough power to cut foamboard and up to 1/8" plywood (firewalls, control horns etc.)
The plywood will most likely take a couple of passes (and lookout for the outdoor grade stuff, the glue they use in that is CRAZY!)
I don't want to dissuade you! I just want to be frank and honest so you know what you might be getting yourself into.