If λ was a non-compact scalar, then functionally integrating over λ produces a delta functional constraint $\delta[\epsilon_{\mu\nu}\partial_{\mu}b_\nu]$ . This means field configurations can be restricted to the form $b_\mu=\partial_\mu\phi$ for a non-compact scalar $\phi$.
First of all, this is literally wrong. Though depending on what OP really means, it could be a careless slip of tongue or genuine misunderstanding.
As long as $\phi$ is twice differentiable, Bianchi identity always holds and $\epsilon_{\mu\nu}\partial^{\mu}\partial^{\nu} \phi = 0$ identically. Compactification or not has no consequence on derivatives, where only an infinitesimal neighborhood of $\phi$ is concerned.
Now, I know that Tong specifically says he wants to relax Bianchi identity in order to allow for winding. That is his slip of tongue. What he really means is not a compact $\phi$ winding around the periodic spatial direction. Rather, he wants to introduce instantons, which are singular configurations where the winding number around the spatial direction changes along the time axis. An instanton in (1+1)-D Minkowski spacetime is topologically the same as a vortex in 2D Euclidean space, and that is where the meaning of "winding" gets mixed up.
The supposed modification to Bianchi identity reads $$ \epsilon_{\mu\nu}\partial^{\mu}\partial^{\nu} \phi = 2\pi \sum_{i} n_i \delta(r - r_i), $$ where $i$ labels the individual instanton, $n_i$ is its strength and $r_i$ its location in spacetime.
Go to imaginary time, and discretize the 2D Euclidean space on a square lattice. The LHS $\epsilon_{ij}\partial_i\partial_j \phi$ is now the circulation of $\partial\phi$ around a plaquette, and should be an integer multiple of $2\pi$. This fixes the periodicity of the corresponding Lagrange multiplier.
If anyone knows of a way to argue for this result without going through the lattice, I'd really love to know.