Is this sentence correct grammatically?
If you replace way by thing, then your sentence is grammatical and makes sense:
Pretending to be a beggar on the street was the only thing such a shameless individual as you could do.
This works because you can do things but you cannot do ways. For example,
Begging was the only thing (that) she could do.
?Begging was the only way (that) she could do.
However, you can have ways. So your friend's suggestion:
Pretending to be a beggar on the street was the only way such a shameless individual as you had
is grammatical and makes sense