When we flew from the USA to the UK in 2017 we used Iceland Air, and we each had two suitcases and a carry-on, and there were no extra charges. I don't know if they still do that, but you might check. It'd fly you into Glasgow, and then you could take a train/bus/ferry to Belfast. (You'll never get on one of the regional hoppers with five suitcases.)
The Daughter flew from Glasgow to the USA on West Jet with five, count 'em five, bags in 2015. There were extra fees, but they were not at all bad. Of course, she had what was basically a bouncing flight - Glasgow to Halifax, change planes. To Toronto, change planes. To Calgary, change planes. To SeaTac, change planes. Then on home. (Hey, student, limited budget, cheap flight!) Not fun, but the she and the bags all arrived at the same time.
You know, that might be an option for you - shipping from London to Belfast. Probably cheaper than extra bags on a plane. There appear to be several services.
No chance you would be driving London to Belfast?