Hello,
to say the truth I don't know if a constitutional complaint (Verfassungsbeschwerde) has a suspensive effect. If it does, you can't be deportet before the Constitutional Court (Bundesverfassungsgericht) has decided on your case. If not, you can be deported. There are of course other methods too which your lawyer can try under circumstances: urgent applications to the local administrative court to stop your deportation temporarily. Your lawyer has full information about your legal situation and can decide if such an appeal is possible / makes sense.
Constitutional complaints are rare and not all lawyers make it. I hope your lawyer is experienced enough to make such a complicated thing and anyway ask him (or her) how much protection this complaint gives you in the moment.
Good luck!