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This bug boils down to the following more elementary problem.

sage: D.<Y> = QQ[]
sage: Dbar = D.quotient(Y-1)
sage: one = Dbar.one()
sage: one
1
sage: D(one)
Traceback (most recent call last)
...
TypeError: unable to convert 1 to a rational

A workaround for your use case is as follows.

Instead of

sage: Bbar.lift()

use the following

sage: Bbar.apply_map(Dbar.lift)
[1 0]

You can check that it lives in the correct matrix space:

sage: _.parent()
Full MatrixSpace of 1 by 2 dense matrices over Univariate Polynomial Ring in Y over Rational Field