The stipulation for F1 and TPS are independent and exclusive. Having TPS does not give you the liberty to work outside the conditions given by F1. Now this is only applicable if you still want to maintain F1. Since you have TPS, you do not need to maintain F1. Working under TPS is legal. However once you use TPS EAD you will lose F1 and be on TPS only. This technical violation of F1 however does not accrue illegal status.

So basically want to play by the rules and maintain F1, do not use TPS EAD. If you don't care about maintaining F1 and only TPS status is ok with you, go ahead. If you want to risk it and use TPS EAD, maybe not tell DSO. Shouldn't be a problem until the DSO finds out because only the DSO has the authority to terminate your F1 status in this situation.