The administration’s hand have been dealt. There is nothing, legally, they can do if the unions reject the TA, to prevent a strike. It would be up to congress at that point if they want to prevent a possible strike.
There is a difference between what administration’s legally can do and what often happens. If the union is told they aren’t released to self help, even though they have jumped through the legal hoops, they would then need to sue in Federal courts. This takes time and money.