yup SH is the way forewards.
Trek 1200/1.2 should be under budget, but has a reputation for being a bit harsh (never tried one but thats what i was told)
Cannondale R600/CAAD5 is smoother, but gives away about half a lb to the trek.
I bought a 2003 (i think) R500/CAAD4 in yellow with a carbon fork for £300 in 2005! Bargain of the century as they still go for £350 on ebay!!!! Added shimano R550 wheels secondhand for £35 which are a lot lighter than the OEM ones, and conti GP4000 tires and 50g tubes. All in Ive spent £450 in the last 10k miles on it!
OEM wheels are invariably heavy and harsh, so budget for some SH aftermarket ones (shimano R550, fulcrum 7’s, etc) , or keep looking for one thats already got them as they dont seem to influence ebay prices that much.
Avoid………
1″ headsets
8 speed