Yes, I agree with mangale. F1 can work with the DHS authorization and that is what it is mention in F1 social security card also. You can take an example of economic hardship work permit. F1 can work more than 40 hours by applying economic hardship. If working more than 20 hours violates F1 status then work permit from economic hardship should also violates F1 status but intact that's not the case. I am not a lawyer here but just reading several post and TPS info from USCIS I got an idea that TPS and your immigration status is two complete different path. You can enjoy both F1 and TPS benefits if you are able to maintain F1 status.

Maintaining F1 :-
Taking full credit hours (12 credit hours for undergraduate) and 9 (credit hours for master)
Working 40 hours with TPS EAD (As this is a EAD that is DHS authorization). Consult your DSO before working on EAD.
Applying H1B from F1