Not sure why the sensitivity of the pickup would be an issue ... if your intonation is out (which is will be to start with) then the last thing you'd want is to have that 'disguised'.
In truth, it doesn't really matter that much what your first DB is. If you take to the instrument, then chances are you'll be planning your first upgrade within a few months. 😂
My first was a £300 no-name Chinese DB (3/4 size). I had an introductory lesson with @Jake Bass to make sure that I had the basics right - how to hold it, what height to set it to, how to press the strings without straining my wrist, how to pluck properly, etc. - and played my first two DB gigs on it at local festivals with a steampunk folk outfit.
Before the Summer was out, I had traded up to a £1000 Zeller which (10 years later) I still gig regularly with my rockabilly band. If you're anywhere near Burnham (Slough) tonight, come along and have a chat. We start at 7:30pm.