It is not こと performing two functions, but simply the fact that last とwhen listing things(in “and” function) can be omitted. (and it is generally the case in casual speech)
AとBと…N(と)
Since どちら/どっち are used mostly with two choices, it means that in this case the second と can be omitted.
AとB(と) You can use comma instead “AとB、”
I answered with とどちら because を is already there and this answer was accepted. But another accepted answer apparently is とどちらが. Is that actually valid? How can you have が and を back to back like that?