Trying to do an airside tansfer when you don't have an onward boarding pass for T4 is a complete non-starter no matter what.

Next you say you are coming in on BMI? Where from? UK&I Common travel area or elsewhere?

Either way, get the flight down to the heathrow express/connect platforms, then take the free train to T4.

In the morning, I'd get a free bus round from T4 to T5 to aviod the shuffle into the central area from 4 then back out to 5.