Have you tried calling Doncaster while you are still on the ground?
As a bit of an aside, I've just looked in the AIP to see what it says (I'm having a very dull day at work!) and it strikes me as the most colossally overengineered bit of airspace for the sort of activity that goes on in the area.
But your aerodrome looks to be firmly located under the EGCN CTA (the bit labelled CTA-1 in the chart on AD 2-EGCN-4-1). If departing from 05 and carrying on in that general direction and staying below 1500' ALT it looks like you could well be outside CAS. Head E to the Trent (below 1500/2000) and as you allude to, you're going to stay in class G. So "Nyahhh" to the controller!
More seriously, especially if you can't get good 2-way comms on the ground, it seems odd that there isn't some agreement or procedure in place to allow aircraft that want to transit the CTR/A to get airborne and up to a decent level in order to get the call in. Perhaps 1000' ALT and staying E of a line drawn N-S through Sandtoft - I've worked with something along those lines in a similar situation.
If I'm permitted to drift off the original point a little (told you work is dull today)...I wonder if the reason for the seemingly unnecessarily complex airspace is an attempt to mollify some of the multitude of objections that are almost routinely raised against any proposal to change or establish a bit of airspace. There surely should be no problem with class D airspace for VFR traffic in a situation like this. And maybe the objections are because controllers have a tendency to treat the airspace as 'theirs' and try to protect it from aircraft whether there's anything significant using the airport or not (I'm not having a go but I've worked with people like that)? I've never really understood that view but it was prevalent in the past. It always seemed to me that the CAA, along with establishing airspace, should have had an obligation to ensure that all users were granted fair and equitable access.
Drat, something's just arrived in my 'In' tray!