When can I "do, do" something?
Solution 1:
The use of do with another verb is for emphasis. It doesn't matter what the other verb is.
We do want to invite you. It's just we don't have the space...
We do love you. But this is for your own good.
We do clean our car. But we drive through mud every day too.
There's no reason that do can't be the verb which is emphasised.
We do that → We do do that.