Bluetooth is designed to be a "weak radio" technology.
So one must expect the signal to be...
weak.
Since Apple designed the current case for the Mini (starting in 2010, was it?), users have complained about bt problems.
Doesn't bt run in the 2.4ghz band (approximately)? If you have a lot of other wifi devices connecting at 2.4ghz, perhaps the other signals are "getting in the way". (I could be wrong about this)
One way to "improve" the Mini's bt performance is to get a 3rd-party USB bt "adapter". Numerous ones are available, but not all work with the Mac, you're going to have to do some research.
And then get a USB "extension cable" like this:
Amazon.com: usb3 extension cable 3ft
www.amazon.com
Plug the adapter into the extension cable.
Plug the extension cable into a USB3 port on the Mini.
"Loop" the extension cable around so that the adapter end is closer to you.
That may work.
Be aware that you also may have to put in some commands in terminal to make the Mini "recognize" the 3rd-party adapter "first". See this:
I wonder how well it would work to call them about it and ask them to send a trackpad since those obviously will work while plugged in. This is the only fair solution I see since they sold us all a device with faulty bluetooth connectivity. There's 16 pages of evidence here, I don't see why...
forums.macrumors.com
I don't use bt, myself.
I use a WIRED keyboard with a USB port, into which I plug the unifying receiver for my Logitech mouse. No connection problems at all that way!