Why British Forces in Libya Were Not a Shortcut to Cairo

Could Britain have used its forces in Libya to overthrow Nasser with a smaller invasion?

No. The Libyan garrison was too small and would itself have required major reinforcement.

Too few British troops were stationed in Libya to overthrow Nasser alongside a smaller invasion force. Reinforcing the garrison would simply recreate the mobilisation problem that the alternative was meant to avoid. Geographic proximity did not supply the combat power or logistics needed to reach Cairo. Any serious advance would still have required a much larger force.