Insulate space above ceiling in 3 season room

The links in the other answers are very informative!

In answer to your question:

So: is the fiberglass a bad idea?

The answer is yes, at least in the exterior walls.

In answer to your question:

If so, what should I do instead?

You have a bunch of options, none as cheap as simple fiberglass batts. My personal recommendation would be rigid foam on the exterior walls and ceiling with fiberglass batts on the newly built walls. Pricey but effective.

Essentially, you're going to be building a little tiny insulated house inside the barn without the need to weatherproof it. At the risk of losing a few square feet of floor space, why not just build the room free-standing in the corner, with maybe a couple of framing attachments (e.g. longer ceiling joists) to the barn walls? That way you can do a simple wall with sheetrock both sides and fiberglass insulation in all four walls and the ceiling.

Actually, since you are primarily concerned with heating conditions, the vapour barrier goes on the warm side of insulation, meaning right up against the sub floor boards. This is to keep moisture from the house from condensing in the cooler insulation. Outside moisture would not condense in the insulation as it should always be warmer than the dew point of the outside air, though in the UK it may often be a close call.

Loose fill generally provides a more complete insulation job, filling the whole space completely. Batt insulation is much easier for DIY work. Even if you access from below, loose fill may be possible. You could actually install loose fill by cutting a hole in each joist bay, instead of removing the ceiling. The main problem is you have no way to verify the void was completely filled. It is highly dependent on the skill of the installer. It also negates the ability to place a vapour barrier, but depending on the flooring system, it may not be a serious issue. In your case though, it is probably an issue.

If any recessed lights are in this space, if they are not rated for placement in insulation, they will need to be replaced. If you go in from the ceiling, it will be hard to get a good vapour barrier. Just do the best you can, sealing it well to each joist. You'll have to allow the joists to be part of the barrier. It may be worth spraying them with a quick latex primer coat to help make a better barrier.