Often times we blame the far end device for the failure instead of the local device in your case the keyboard and mouse. The system side is not your issue it’s the limits of your keyboard and mouse as there signal is not as strong.
Depending on what your door and walls are made with they can attenuate the signal reducing the distance in addition there can be a lot of radio noise which interferes with your given signals in your surroundings. Like other computers and TV and radio devices which are leaking into the BT bands.
A simple BT repeater is all you will need placing it closer to the system in line of sight where you are using your keyboard and mouse. Any BT repeater will work here as long as it matches your keyboard and mouse needs. Review the spec sheet of the keyboard and mouse to figure out what you need. The system doesn’t need to be altered to address the limits you placed on it.
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