Don't do it.

More realistically, you can't do it within budget.

A cheap UST projector of any quality is right up near your budget. Then you really must have a tab-tensioned screen if you want a retractable screen.

If they couldn't tell you why, then you were dealing with someone who didn't know projectors and projector technology.

The reason is simple: Non-tensioned retractable screens aren't actually flat. They have waves across the surface. Sometimes minor at first, those waves in the material get worse over time. Kind of like your curtains at home, or a shower curtain. A non-tensioned screen just doesn't sit 'flat', and it can't.

Why does this matter?

Because you aren't talking about taking a projector and putting it on your ceiling from across the room, you are talking about putting it a few inches off the wall and projecting a geometrically accurate square onto a NON uniform surface. This will create significant distortion (inches of it) in the image that you project. The image won't be nice and square, but will be all wavy.

Add to this that ultra short throw projectors are a beast to setup accurately. They have millimeter accuracy issues and any placement problems can result in an image that isn't at all square to the screen or able to fill the screen the whole way. I would expect, after you got good at it, that it would take about 30 minutes to accurately setup the projector to be reasonably accurate.

Options?

You could use a standard throw projector at the opposite side of the room and could live with a regular ambient light rejecting screen. It will look good at night and okay during the day. It would still likely be a tab-tensioned screen if it is a ambient light rejecting (ALR) screen. Those screens often cost $3,000 by themselves. They aren't inexpensive.

A standard screen is an option, just a manual or electric non-tensioned screen. It will require you to get good curtains over the windows and control your ambient light to get a usable image during the daytime.

At best, these look 'okay' during the day. You get nice size. But, the image with any ambient light will not compete with that of a LCD or OLED flat panel.

Frankly, this room isn't a home theater space and the physics you are fighting are less than ideal. The location, being over a fireplace, is also just about the worst it could be in a home.