If I had entered into a USD 10mn pay SOFR, receive 3M LIBOR swap with a 5yr maturity, I would have had a positive NPV of about 80k by the beginning of March due to the massive drop in SOFR (1.55 to 0.01) while 3M LIBOR fell from 1.83613 to 1.21563. I understand the mechanics behind the pricing. However, I am looking for a risk measure to use in this context. DV01 is meaningless due to the small value and BR01 (change in value for a 1bp shift in the basis curve) is not available for this type of basis swap. What would I best use to describe the risk for this instrument.
I will try to make a more general suggestion that doesn't depend on SOFR, EONIA, LIBOR, cross-currency basis, etc, but applies all all linear interest rates products. Sorry if I may be digressing.
You have a number of observable instruments that you use to build your interest rate (multi)curves. In general, these are the hedging instruments that you would use if you wanted to change your interest rate exposure. Or at least the instruments that you observe and mark.
For each of these instruments, you calculate the delta to this instrument by bumping this instrument up and down and refitting the curve. These risk measures will give you the best P&L explain for linear products. E.g, if you know that you had $\\\$n $ delta to some interest rates future, and that future has moved $m$, then you had $\\\$mn $ P&L from that. (I won't go into theta and gamma and cross gammas here, but it's all do-able.)
But these risk measures are less than perfect for risk reporting. For example, you may calculate directly the sensitivity to a rates future (because that's what you'd hedge with), but you prefer to see on some risk reports the sensitivity to a swap rate in that tenor. It's best to use inverse Jacobian than to calculate these sensitivities directly. Calculating the sensitivies of your "reporting" instruments to your "hedging" instruments, then use matrix multiplication to translate the vector of your book's "hedging" sensitivities into its "reporting" sensitivities. If you further want to see the sensitivity to a parallel shift in all swap rates (ie dv01s to SOFR swap curve, LIBOR swap etc) (or further to $n$ standard deviation change in the first 3 principal components of the swap curves, etc) you can quickly estimate those by summing up the sensitivities to the corresponding swap rates, or you can use inverse Jacobian again.
A good tutorial on using inverse Jacobians to transform interest rate risk measure is here: https://www.closemountain.com/papers/risktransform1.pdf
In other words, I think it is better not to look directly for the summary numbers (SOFR swap curve dv01, LIBOR swap curve dv01), but to build them up from the sensivities to the curves' building blocks.
In your concrete example, imagine hypothetically that instead of USD LIBOR v SOFR, you had a cross-currency basis swap - USD LIBOR v GBP LIBOR or EURIBOR or JPY TIBOR... How would you react if someone told you that you can only see the net dv01 added up across all currencies and tenors, and that the sensitivity to the difference between USD and other currencies interest rates were "not available"?