Let me do this.
Going out on a limb here, but would a German ever say “Lassen Sie mich das machen”? I want someone to “let me” “do that” which puts each verb with what I refers to. I will humbly accept a denial, if such is the case.
I would even say “Lassen Sie mich das machen” is the more natural option here.
“Lassen Sie das mich machen” is possible but overly emphasizing the “mich”. It’s like “Let me be the one who does it”.
The former can be interpreted both ways, depending on whether the emphasis is on “mich” or on “machen”, but it’s still the more natural way for both, in my opinion.
Both sentences are totally fine.
Your proposal “Lassen Sie mich das machen” sounds more natural, though.
You can argue that each sentence has its own use case:
Situation 1:
You see someone struggling with a task, and you want to take it over from them (for example, a grandma carrying a heavy bag):
Lassen Sie mich das machen.
Situation 2:
Your boss has two tasks to assign, and he gives task A to your co-worker and task B to you. But you’d prefer task A. (Or a mother assigns her two children two tasks).
Lassen Sie das (task A) mich machen.
It doesn’t even have to be two tasks. It could be one prestigious task that was given to a co-worker, and you ask your boss to give it to you instead.
In the end, it doesn’t really matter though. Both sentences are fine for both situations.
See also:
If there is an emoji for “huge sigh of relief” imagine it inserted here. Grammar always matters, and it matters really a lot in German, which is why I love the language. So, thank you for once again giving me ground to stand on.