I have nicely calibrated the SABR params for the USD 3m Libor swaptions which matches the market very well (after incorporating the correction due to Obloj - Fine Tuning the Smile paper). I am extending the pricing model to cover mid curve swaptions. Here I express a mid curve rate such as 3m1y1y as a combination of the 3m1y and 3m2y rates. Now, in order price the midcurve non ATM options, I need to get the distribution of 3m1y1y. Is it possible to approximate the midcurve distribution as a SABR distribution as well and if so can we express (approximately) the midcurve SABR params in terms of those of the 3m1y and 3m2y ? I have not encountered such a result before and I thought it might be an interesting approach. One piece if information I have is the midcurve ATM normal vol. From this I can back out the market implied correlation between the 3m1y and 3m2y, since I already have the respective rates and normal vols. So any suggestions on how to mark the midcurve skew consistently with the swaption skew will be helpful to me. Looking around in the forum, I found Martinghoul's GS paper which describes how to price the midcurve ATM swaption, but it doesnt talk about the skew.

the midcurve atm vol and the correlation (also the atm correlation) say nothing about the skew. One way to build a midcurve model consistent with SABR for the underlying is to treat the midcurve as a spread options, in fact 3m1y1y can be treated as a 3m forward 2y-1y spread options, and given this you can get the marginals from respective SABR and use a model (perhaps a copula) to develop the spread option price.