Would a 'better' driver make any difference?

as long as it's appropriate to you in terms of loft shaft and any inbuilt bias the simple answer is yes

I wouldn't budget beyond the bottom end of you listed numbers - even buying new I haven't paid more that £99.99 for a driver - so take advice from who ever is teaching/guiding you, try a few other peoples drivers at the club over time, and take it from there.

where abouts are you playing (when the snow clears obviously!)?