You're correct in thinking the brakes should work with any shifter. You'd also need a
8 speed triple brifter or you could go down the
bar end shifters and aero lever route.
You'll probably need a new road front derailleur to work with the road shifters as they have different cable pull ratios.
The drop bars will extend your reach fairly significantly unless you spend all your time riding on the flats, so you'll need a new shorter stem if you're not comfortable with that.
Add new bar tape and cables and you're looking at something like £160 with brifters if you do all the fitting work yourself. It could go down to slightly less if you're happy with bar end shifters.
If you're happy doing that there's no reason why you can't, but personally I'd get some bar ends and put the money towards a new bike. (Triban 3?)