But twins look so bitchin' 
My 1275 (now 1293) runs a HIF44/HIF6 on an alloy water heated manifold (from Minispares), a MED stub stack and ITG filter, All of this fits without hitting the firewall. I run a BBW needle and LZX2085 damper for what it's worth.

It looks like there are velocity stacks taking up the available space, if it were me I would buy aftermarket K&N pancake filters and skip the stacks.

You are possibly going to have the same issue with a HIF. A picture of your bulkhead underneath the carbs would help.
In the early 90's when Rover used the 1.5 " 38mm HIF on the Mini's they also moved the engine forward by re drilling the sub frame mount holes so the float bowl would clear the bulkhead.
Most owners that cannot keep twins in tune i have found are trying to do so with worn throttle spindles, once you set up a pair of twins they should need very little adjustment if everything else is set up correctly.
I believe the only factory manifold that clears the bulkhead with a HIF 44mm 1 3/4" (on later cars anyway) is the MG Metro one.
If it were my car i would find a correct 30 degree angle twin 1.25" manifold and re build the carbs 100% if you have not done so and make sure everything else like timing, max advance, vacuum advance, points (if fitted), valves set, plugs gapped correctly before you begin to set the carbs.
Cheapest route and easiest to fit would be a single HS4.

You may be using carbs from an ADO16 car.  The biggest (maybe only) difference to those for a mini is the float angle I believe.

Perhaps the manifold is incorrect also (?)

This can be fixed with the correct float bowl grommets.... (30 degree I think )

Early 90's run of the mill 1275 cars came with HIF38 and was adequate, but I think HIF44 would be better and cover any further tuning....just make sure you have the correct needle for your build/breathing.

I have a 63 Mini Traveller that currently has twin HS2's on a 1275. It has the incorrect manifold/carb (both?) setup as carbs hit firewall and I cannot run even the thinnest air cleaner. Has both top and bottom engine stabilizers and I have even thinned-down the spacer block by 1/8". Still "thumps" at low RPM. Besides, I am sick of trying to synchronize carbs and deal with 2 mixture settings. My question: What is the best single-carb setup to go with on stock 1275? I seem to remember this question being asked here some time ago with an answer of single HIF44 but not sure and I cannot find the earlier thread. Advise appreciated!!