Good that you publicised this incicdent.
If you can't clear from the frequency you were on, it becomes rather like something in the 'rear view mirror'. The priority being to think about what's ahead.
Given the exact circumstances again, I would recommend:
1. Call the Solent controller and obtain the relevant clearance.
2. When (and only when) you've received that, advise that you weren't able to clear with Shoreham, so would he please do so for you.
If you are the sole licenced FRTO in the aeroplane, there is just you and one, maybe 2 radios. On the ground there will be several people and plenty of phones. Don't think twice about requeting that the ATCO gets an assistant to advise Shoreham.
I recall routeing near Solent once - and decided to call them to advise. I merely received a very curt "C/S, Standby". So I did, but continued on my way within 0.5 mile of their airspace. When they finally decided to speak to me, they were somewhat taken aback to be told "I was just going to advise you that I was going to be flying quite close to the edge of your airspace. Which I have been. But not now, so thank you and good-bye. Squawking7000toenroutegoodday"