Some incorrect assumptions here, I think:
need to descend to 1300 feet AMSL (following the recent “take 2” guidance )
Take 2 is useful for infringements of airspace only, not for clearances within airspace.
the ATCO is presumably going to think I’m having a laugh
Nobody is going to think you're having a laugh. You request the route / altitude you'd like. You maintain VFR and clearance of terrain. The controllers permits it, if available. If not available, you could be offered alternative routes / altitudes, or even put int a hold for now. If it's too busy or the controller deems the situation unsafe, you won't be allowed in. If you cannot ask / accept for a VFR clearance then request for SVFR. If the route takes you into cloud or terrain or obstacles, then say unable.
Definitely avoid flying into cloud first and then ask "Can I change to IFR" or declare "Pan" (if you have no instrument experience) as it'll take much longer to arrange and cause you a lot of pain in the process.
I’ll need to co-ordinate another descent with him, being careful of the overflying and glide clear rules. . Now I'm out over the Thames at 1300 feet, super. Is this really a goer?
Of course! If you have to ditch at sea, it is better there than into buildings.
Is it not so fraught with hassle that it will greatly diminish the chance of getting a transit at all, much less a timely one?
Depends on the traffic and the weather really. Check the visual approach chart if there are any published/standard transit routes which could be less hassle. If nothing published, then you're free to ask whatever, within reason.
If we don't take part in protecting GA access to infrastructure, the smaller aerodromes will be built over, the larger aerodromes will have exorbitant handling fees and awkward procedures, and airspace will be over-classified to Class A everywhere.