Humberside would not refuse to provide a service. They may be too busy at times to provide a Traffic Service or Deconfliction Service, but a Basic Service is certainly always available (when they are open). If they think you should be speaking to a more appropriate unit, they will say so, ie. Waddington/Coningsby/Linton etc.
As has already been said, Doncaster isn't a LARS unit, but Humberside is, so in most cases (unless you wanted to transit Doncaster's airspace) Humberside would be the appropriate unit to speak to.
Just to add, re workload, at most times during the day a second radar postion is available, so if it gets really busy, rather than refusing you a service, Humberside Radar would open the second radar position, so you might be talking to 119.125 (their first point of contact) or they may ask you to re-call them on 129.250.
Hope this helps.