Now that the UK LAPL is a wholly UK licence restricted to UK airspace perhaps they may allow the use of a Medical Declaration.
Sorry that I mislead you when stating the above but yes you can.
From the CAA website:
A medical declaration (from 25th August 2016) is an affirmation of your medical ‘fitness to fly’ and may be used to exercise the privileges of a:
• UK Part-FCL Private Pilot Licence (PPL) to fly UK (G) registered Part 21 and non-Part 21 aircraft;
• UK Part-FCL Light Aircraft Pilots Licence (LAPL) to fly UK (G) registered Part 21 and non-Part 21 aircraft;