Hmm, maybe I am misunderstanding. I thought the idea was just to use Botanical Gardens as a walk through. If so, I am not sure I would pay £4 just to walk through as a short cut as it is not that much of a short cut and if your child is on bike or scooter they are not allowed to bike or scoot through. In fact may not be a short cut at all.
If coming from Rail Station on foot to Lammas Land, it is left out of Station Rd, cross rd, right into Brooklands Ave, cross rd at end, follow cyle/foot path over fields and 2 foot/cycle bridges to Lamas Land playground and paddling pool.
Through Botannical gardens route - right at end of Station Rd, cross road, back track to entrance of Botanical garden, walk through, exit garden and walk left to get to same crossing of rd as at end of Brooklands Ave and then as above.
But as a trip in itself the garden is very nice.