Do you have a sense that getting the time-sensitive thing done without the long rest is possible at all? Like, had you avoided some of those combats? Curious about how that was structured.
The time-sensitive part included several days sailing each way. It wasn't a "per-hour" type of thing
And the DM was using random encounters in an old school way - had us rolling a die every so often. We had two encounters on while sailing. One we availed, the other was with a faster ship that all of our attempts to avoid failed and was a forced fight.
The other fight was traveling across an island, up a set of lava flows that were our only way to get to where we were going. We tried to negotiate out of it but they were Minotaur worshipers of Mogis the God of Slaughter (we're in Theros) who recognized the vestments of our cleric of Purphoros and attacked.
So neither of those do I think we could avoid, and we had tried.
The seeming trigger for the cyclops fight was removing the burning incense from the shrine of Mogis, don't really know if that was avoidable but there's a good chance he was expecting us to retreat. There were caves not too far off that were too small for it - I think he was expecting to go there, but our cleric did a martyr "you all run, I'll stop it" for various RP reasons, and then no one
All of that said, the DM had been hinting to the newer players that we should rest. And with our casters out of slots a short rest would help with HPs, but my rogue would have been the only one at full fighting trim - the barbarian was out of rages, the paladin was out of slots and all but 5 lay-on-hands, and the cleric and the wizard had a single 1st level slot each at the beginning of that.
Later we ended up resting in the caves and he put out hints of riches deeper in them, but we decided to follow up on our time-sensitive mission instead.
This part of a larger campaign - if we take too long on this it will have repercussions but it's not like it is a one-shot where we are seeing if we can succeed on this particular adventure.