You've identified a need, due to your gearbox, and anything is possible; the question around it is what is practicable for your situation.
Remember that unlike the momentary action of a Headlamp Flasher, the Dip Switch has to be a Changeover Switch.
Unless you're going to 'over engineer' it with a relay, it needs to be capable of taking the current of 2 headlamps, depending on what headlamps you have fitted.
It needs to be fitted in a position where you can readily reach it, perhaps frequently if you're the sort of person who jumps to Full beam as your default lighting, some are, some aren't.
You'd need a SPDT (Single Pole, Double Throw).
If you have halogen headlamps, it'll need to be rated around 15A.
If you've on LED headlamps, it should require far less.
Latching push button are quite rare in the heavy current spec you require, so you'd probably have to get a toggle switch, but they can be quite firm to operate due to the heavy current construction...
Therefore, you might have to go the relay route, which wouldn't be my preferred route since it has the potential for a relay failure to leave you with headlamps stuck on full beam, but practicality might dictate that solution.
I did a quick search and found that the Headlamp Dip Switch for the Jaguar E-Type is a Toggle, but no retailer I found actually gave a tech spec.
You might like to make enquiries.
