"Let's/We let A be a variable"

You should simply write and say

Let A be a variable

and the audience will understand what you are conveying.

Let us let A be a variable

and

We let A be a variable

are fine but unnecessarily long.

I'd use "Let A be a variable" in speaking, too.

"Let's let A..." is correct grammatically, but sounds unnatural.

I wouldn't use "Let us A...", it isn't correct.

"We let A be..." is correct, but I insist that the first option is the best one.

Let's let A ... is grammatical. Let us A be... is not.

I would say Let x be variable and I would avoid using A. The reason is that a is already an overloaded letter in English and it's not immediately clear whether you are about to introduce a singular noun (e.g., Let a cat out of the house.) or the name of a variable.

Another useful phrase would be Let lambda equal as it's likely you are about to introduce the definition of a variable. If you are talking about a variable that's in the middle of an equation, you could just use a declarative: mu is variable or c is constant.