If you can put the new bolt in and it spins freely you can try just putting the next size tap in there. HD sells tap sets as well as single taps so you should be able to find one with no problem.
You basically ripped out all of the threads and since a starting tap is tapered you should be able to do it without drilling it out.
The problem would be the tap handle since you say the space is limited.
Spray it with lube then 1/2 turn in 1/4 turn back. Keep doing that until the tap goes all the way through, then spray some more lube in after to clean it out.
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