The two models (or the two families of models) are NOT compatible with one another. The recommended practice is to assume an LMM and then seek the swaption prices under such a model.
Correlation across rates of various maturity is much more important for swaption than for cap/floor, because the swaption payoff cannot be separated into individual expectation terms (in other words, when swaption expires we decide on whether to exercise based NOT on the sum of "swap-lets" - there is no such thing, but on the swap as a whole. cf. cap/floor payoff, which are nothing but sum of payoffs of individual caplet/floorlet).
Now back to LMM. We pick a set of forward LIBOR to fit. Each forward rate F(t,T1,T2) is a martingale under its 'natural' probability measure using P(T2) as the numeraire. However if we pick one single P() as the numeraire for all forward rates, most (except for one) rates will NOT be martingales. Thus we also need a formula for the dynamics of F(t,T1,T2) under some other measures. With this formula we can use MC pricing.
Bottom line: The rates "look like" tradable assets
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