Need a recommendation but not sure what I'm asking for

I bought a cheapo and learned some lessons about what I wanted. I'm not 100% sure what the terms are for the stuff I need to look for, or even if I can find them on projectors that I'm willing to pay for. This is for outside, and my use case doesn't make sense for me to go far over $300 (I'm in the US), and even that's going to make me frown a bit while I'm paying it.

I have a 100 inch screen that supports rear projection. Rear projection is a must because of placement. I have 8-10 ft. behind the screen where I can set the projector. Filling the 100" screen isn't a requirement, but a larger picture is preferred.

I'd like to be able to place the projector offset and possibly below the center of the screen. I understand the loss of resolution and pixels with keystone. What is more important to me is to have a rectangular picture where the whole picture is in focus. The current cheapo projector I have can do keystone one direction, but still focuses on the plane with the projector, so if you have the keystone adjusted (manual) then only part of the image is in focus.

I'd like it to be able to use a Bluetooth speaker well. I don't know how to describe "well" except that the current one is bad and I want better. I know that's not much help, but this is the least important part and I can find workarounds to this.

TIA.