There are two quick and dirty solutions. First would be to disaggregate series B to weekly values (R package tempdisagg is great for that) and then do a usual model. Second aggregate series A to monthly frequency, do a forecast and then use disaggregation on the forecast.

The more theoretical approach would be casting problem to a state space model. There are a lot of literature on state space model approach when the dependent variable is observed at lower frequency. It usually assumes that the low frequency variable is really a high frequency variable observed at low frequency periods. You can make the same assumption and then reverse the methodology. Unfortunately I have not seen something similar being done, but I did not look hard enough.

There are two quick and dirty solutions. First would be to disaggregate series B to weekly values (R package tempdisagg is great for that) and then do a usual model. Second aggregate series A to monthly frequency, do a forecast and then use disaggregation on the forecast.

The more theoretical approach would be casting problem to a state space model. There are a lot of literature on state space model approach when the dependent variable is observed at lower frequency. It usually assumes that the low frequency variable is really a high frequency variable observed at low frequency periods. You can make the same assumption and then reverse the methodology. Unfortunately I have not seen something similar being done, but I did not look hard enough.

Concerning midasr, I can say that it was designed to work when the dependent variable is observed at the lowest frequency. The reverse situation was not seriously considered.