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Look around for the first generation di2 hydro shifters - rs785 - pair with a grx or long cage di2 mech. depending on frame you can probably get calipers for 120ish. as @hippy said, waiting for parts to show up over time is cheaper - otherwise things like junction boxes and batteries eat up a lot of the budget. also, some good deals to be had on etap axs - with just shifters and 1 rear mech it might be cheaper - i believe you can swap out the jockey wheels to run a non-flattop chain and cassette (and avoid the expense of a freehub).
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How much are the 785 levers going for? (I see a pair on ebay for £99). I very well used pair sitting in a drawer here that I replaced with GRX on the Tripster.
I also have a short cage RD and potentially the scraps of a couple of other Di2 RDs that met with cars, etc. I salvaged one Di2 TT shifter by stripping it out of its case and taping it up but the other one's cable broke right up near the buttons and I've not bothered to crack it open to rejoin cables - you're welcome to that @swedeee but it's probably more trouble than it's worth. It really depends on how 'budget' is budget.
I've probably binned/donated more 11spd chains and cassettes than most people get through in their lives.
a genuine which groupset question;
what is the cheapest way to get an electronic hydro disc groupset at the moment? i am aiming to build up a winter bike but it must be electronic shifting + hydro disc. fine with mid-range calipers and crankset etc.. it would likely be for a kinesis 4S disc roadset. it could be 1x11/12 if that makes it cheaper/easier to get hold of. aware that everywhere seems out of stock of everything atm