Nate's choices were to climb the ladder and hope to have enough time to sprint across the top, or to get bogged down in the foam blocks in the Pit and then have to navigate the Ledges, which were just as time consuming. It was probably in his best interest to climb up into the Chamber, but time just wasn't on his side.
And I'd like to point out that even though I said "time wasn't on his side," I do not mean that Lissy wasted the time. She completed her third room objective with 1:55 on the clock, which meant she kept an average pace throughout her stay in the temple (as compared to other Season 3 frontrunners). What I am saying is that Nate and Lissy had one of the longest paths to the Tomb of the Headless Kings, since they were the only team to be forced to complete both the Chamber and King's Storeroom en route. Unfortunately, being forced to complete both of those rooms used up time that could have been used to escape the temple.