Familiar problem, though my context is slightly different — I've been dealing with the same thermal management headaches in my motorhome for years, and the physics is identical regardless of whether it's a timber-clad box on wheels or one bolted to a patio.
Insulation alone does almost nothing if your glazing is unaddressed. A single south-facing window can overwhelm even 100mm of Kingspan during a decent British summer. External solar shading — proper blinds or overhangs — makes a far more dramatic difference than adding another layer of insulation ever will. Interior blinds just trap heat between the glass and the blind, which is largely pointless.
For active cooling without mains dependency (or with minimal draw), I've had good results running a 12V compressor-based unit fed from a Victron MPPT system and a Fogstar lithium bank. Obviously that's motorhome-scale thinking, but a garden office could absolutely support a similar setup with roof-mounted panels.
The other overlooked factor is thermal mass. Timber garden offices heat rapidly because they have virtually none. A small amount of dense material inside — even a slate or concrete floor — acts as a buffer and stops that horrible spike you get around 2pm.
Ventilation positioning matters enormously too. Low inlet, high outlet, ideally on opposite faces to create cross-flow. Most off-the-shelf garden offices have vents positioned for rain protection rather than thermodynamics.
What orientation is your office, and what glazing spec are you working with? That'll determine which solution gives the most bang for your effort.