"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.

You're a teacher. Communication is possibly more important than correctness. For this reason, I would use:

Let A be a variable.

Also, this is posed as a command to your students, which is applicable in your situation.