Jim Macklin wrote:

An IFR clearance can certainly be given to a departure in
Class G that is under or abuts. You can often contact ATC
and get a void time, either over the telephone or by radio
if there is a near-by facility.
The clearance would be something like ...
ATC clears N6662U to enter controlled airspace on heading
270 at 3000 feet. Cleared to the XYZ airport, via . Contact
128.975. Clearance void if not airborne by 1730, time now
1715.


Happens all the time.

But, check the airspace issues at the OP's airport.