Step one is to do a heat load calculation for the FROG. If the folks who came out didn't do one (They might call it a "Manual J calculation"), you can do it yourself in just a few minutes. I like this calculator:
HVAC Load Calculation - Maunualj - Whole House Loadcalc
Don't overlook that garage underneath--it could add a lot to the heat load of the FROG, especially if there's no insulation under the floor of the FROG.
We haven't talked about heat-- is the present system for the house doing an adequate job of keeping the FROG comfortable in the winter?
FROG has two ducts. Do we know for certain that one duct is a return?
Yes. If there's no return air (through a duct, through an undercut door, through a grill in the door, etc), then neither the heating nor cooling will be effective.
Before even considering using your present HVAC system, you'd need to know if the existing system has sufficient capacity, which is another set of calculations (or, you could base it on experience if you know the duty cycle of the AC system on your very hottest days. If it is running continuously, then it can't handle the additional load of the FROG). To make the temps in the FROG right, you'd need to add a separate zone to your present system IF it has sufficient capacity (so a separate thermostat, mechanical dampers, etc). It's still a compromise, and your system will be running at times when it otherwise wouldn't need to. On the whole--I wouldn't bother with that. I'd just get new equipment.
To restate some already said stuff, you have 5 approaches available:
1) Integrate the FROG into your existing HVAC (heating and cooling) system ductwork. For reasons above, I wouldn't do that.
2) Window AC unit for the FROG. These can be installed in a window OR a newly framed opening in an existing exterior wall (if the FROG has one--if it is all hip roof overhead, then you don't have this option). Cutting a new opening (vs using a window) means you don't lose any of the existing light, and you can mount the unit fairly high (which is best for cooling). Yes, you can buy a special unit designed for wall mounting, but a mass-market window unit will work just as well and will be less expensive. If you don't need to heat the FROG, this is a very inexpensive way to go, and the efficiency isn't bad. As stated--when the unit crumps out in a decade or so, you go get another one and you are done. Very simple. It will be a bit noisier than a minisplit, but it might not be a problem.
3) Minisplit: A deluxe answer and probably the answer most likely in an upscale home. When any part of it needs fixing, there will be a pricey service call. More efficient than a window unit, but the payback in utlity cost would for the minisplit would probably exceed the life of the equipment.
4) Minisplit with heat pump: Like Option 3, but with heat. More $$.
5) Window unit with heat: Like Option 2, but with a built-in heater. These are typically electrical resistance heaters ("heat strips"), so they are expensive to run. If your existing ductwork does the heating job, but not entirely well, this could be an option. Also, if you just need to heat for a month per year, or several hours per day, this might be fine.
You could also use a through-the-wall PTAC unit--which has cooling and heat-pump heating in one piece of equipment. These are common in motels, and cost less than $1000 for the equipment. Look for a quiet one. Examples:
https://www.alpinehomeair.com/viewcategory.cfm?categoryID=248
A FROG can be some really useful space. We had one when we lived in NC, and DW and I spent a lot of time there--puttering around on our projects, etc.