Gatwick overhead transits were previously limited to rotary traffic only but this was recently changed since the CAA will not allow such a restriction in class D airspace. However, most tower controllers are very cautious about such traffic, and of course thr IFR traffic will always take priority. The weather will need to be very good, if not CAVOK and the actual traffic demand will need to be low before we're willing to take fixed wing traffic at the VRPs.
However, if I happen to get the call from my radar colleagues at the right moment (mid to late afternoon usually), such that I don't think you'll have to hold for long at all at the north terminal or southern maintainence hangars, then I'd be happy to take you through.
Be ready to orbit at those VRPs though, and not to infringe the runway extended centrelines until cleared.
Finally, bear in mind that there may be any number of reasons in addition to weather and traffic that may increase a controller's workload to the level where they feel vindicated in refusing.