The answer is no.

In a cancellative monoid left invertible elements are right invertible. If yx =1 then xyx=x and so xy =1 by cancellation. So you are asking to invert a given element in some cancellative monoid.

Take a fg cancellative monoid M not embeddable in a group. Enumerate the elements of M and invert them one by one. Then the original monoid would embed in the unit group of the direct limit of this process.

The answer is no.

In a cancellative monoid left invertible elements are right invertible. If yx =1 then xyx=x and so xy =1 by cancellation. So you are asking to invert a given element in some cancellative monoid.

Take a fg cancellative monoid M not embeddable in a group. Invert the generators one by one to embed M in the group of units of a monoid and get a contradiction.