I actually did the Courier's Rasher before I had double jump and even before I had ~~silkspear~~ clawline, just cling grip and drifter's cloak (float). I was impressed with myself because after doing it, I looked around and couldn't find any video of anyone making the run without at least clawline. But... I also did it in way that is probably considered rather cheesy. I'm sure others have realized it too, it seems pretty obvious to me, but after I did it, I searched around to see if any guides or tip posts mentioned it and none did, at least that I could find. It takes a bit of prep, but if anyone reading this is having serious trouble making it to the kitchen in time, but you just want to get that damn timed mission out of the way, here's a cheesy method that could make it a lot easier for you:
spoiler
I'm assuming you already know the quickest route to take. But here's the thing, you can go through that whole route before hand and kill the enemies that you would encounter along it before even accepting the mission from Tipp and Pill and those enemies will stay dead. Obviously then when you do accept the mission, it's clear sailing. You just have to worry about sprinting enough and doing the platforming/parkour bits and whatnot, but not having to worry about having to kill or evade the numerous enemies along the way which is really what makes the run so damn hard.
There are some exceptions for pre-killing enemies though. As you probably know, there are basically two categories of regular enemy in Silksong: the enemies that when killed will stay dead when you leave a room or area and return, and those that always respawn immediately upon returning to a room or area. The former ones tend to be the larger ones, like all the pilgrim enemies, and fortunately they are mostly what you encounter on Courier's Rasher. But the little ones like the little flies- and mice-styled bugs will be there again when you return to a room even if you killed them seconds ago.
So to prepare, you want to go along the route, probably best to start at the finish line and work your way back towards the start at Tipp and Pill, and kill off all the various haunted pilgrims in the Choral Chambers, at the Blasted Steps there are some judges (brass mechanical, shield-bearing guys) you might encounter so it is good to kill them off, more flying pilgrim enemies in the vertical chamber at the end of Shellwood and the pondcatchers (flying spear-throwing pilgrims) along that first long horizontal part of the run in Shellwood. But don't bother with the pond skippers there where the pondcatchers are, don't bother with the driznits (the flying bugs that shoot their drill-shaped face at you) at the blasted steps, or with the pharlid towards the end of the run - those ones will be there anyway when you come back through after starting the mission. But the ones that respawn mostly don't cause any real issue as you're running through and can mostly be ignored.
The only respawning ones that tend to cause problems are the driznits. If you've done the runback for the Last Judge boss, then you've probably already figured out how to avoid them for this run too, as it's the exact same route. All of them can be ignored and ran past so long as you sprint, which you need to do anyway to make the Courier's Rasher run in time.
If during this preperation, where you're killing off the enemies before the mission, you take damage and are close to death, just find some of the little enemies that do respawn, whack 'em to build silk and heal as you normally would. If you kill them and are still low on health, just leave the room and come back. I used the pond skippers to make sure I was at full health before starting the mission but after doing the rest of the prep. What you obviously don't want to do is rest at a bench because that will reset all your prep work.
tl;dr
Kill the enemies along the route before starting the mission, just don't rest after doing so, then run through the mission without encountering them.