My 1st mini was an 850 with an 1100 engine fitted, twin carbs, 731 cam, big valve head and would rev to 7000 rpm.

Sadly the poor gearbox suffered and was being contstantly rebuilt. Short answer is yes it can be done and I think if you read a few old mini books you will find how to do it. I think the gearbox casing may need some grinding to allow for the 1100 crankshaft and to stop it hitting?
Your better option may be to try and find a pudding stirrer fitting for the remote type gearbox. Then you can keep the look and also have synchro on 1st gear.
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1969 Cooper "S"
1967 Mini Deluxe
1973 Clubman Van (the fleet spare)
1978 ex 1275 LS ("Wizard" Eaton Supercharged)

2015 HSV GenF GTS(occasional drive & tow car)
2019 MINI F55 Cooper S