I think I may have screwed up the order of operations a bit
But because you're using an oil finish it's not quite as bad as it seems.
Inside the cracks and other defects the epoxy won't stick as well as it would to bare wood, however once cured drying oils aren't oils any longer so the bond won't be as terrible as one might expect. Long wait for the oil to fully cure though — a month at least!
Another thing to bear in mind is that fills like this can often lock in due to the complex topography of the defect, so many fills couldn't be removed regardless. It just takes a little shelf or ledge in the defect for the epoxy to grab on to and the fill holds itself in very well1.
Will the process of sanding it down flush damage the surrounding oil finish in a way that will be noticible in the end?
It will damage it but shouldn't be noticeable in the end. One of the (few) great things about oil finishes is that they're very repairable, you can do spot repairs and they should blend in seamlessly (see more on this at bottom).
BTW when you do this sanding you'll have the opportunity to see just how shallow the penetration of oil is into long-grain surfaces. The "deep penetration" that finish companies like to brag about with oil finishes is only relative to other finishes which sit on the surface, it's not actually deep at all!
The whole process
There are other ways you could approach this but this is how I'd do it:
Wash out the inside of the defects with acetone or lacquer thinner to remove some of the 'dried' oil (Q-tips or small, stiff-bristled brushes are helpful here). You don't have to be super-scrupulous about this, the more you can remove the better but don't drive yourself nuts if you see traces of oil remaining after the solvent has flashed off.
Gently scrape the surface of the defects with a sharp point to key it. If you've waited for the oil to cure and you want to skip the above step and go straight to this one go ahead, at the end of the day it may not make that much difference.
Just prior to applying the epoxy use a hairdryer, or more carefully a heat gun, to warm the area well. The higher temperature makes the epoxy flow out and penetrate better, ensuring it runs into every nook and cranny. Even viscous epoxy mixes thickened with filler will turn surprisingly runny the moment they touch hot wood (#oddlysatisfying). Note: always do this when using epoxy if the wood is cool or cold if you want to ensure a good bond.
Mix your epoxy and pour or dollop it in. Make sure to overfill! If needed use tape or other things to dam the surrounding area to keep the epoxy from dripping or flowing where you don't want it.
Wait for the epoxy to completely cure2 before working the surface flush. If you have quite a bit of excess and you want to remove the bulk of it before the epoxy has reached maximum hardness do so, but hold yourself back from trying to get it flush or you risk tearing it free from the wood at the edges.
I like to plane the bulk of epoxy excess off, usually using a block plane set for a fine cut although with a larger fill you'd want to use a smoothing plane. Once I'm flush with the wood I might scrape and then finish sand to fully blend the fill in. You'll get much better edges to your fills if you've been patient and waited for the epoxy to fully cure.
Start oiling again as normal. Concentrate on the bare areas of course but you'll probably end up oiling most of the surface again as you go even if you weren't intending to add further coats (which I would recommend actually, three isn't enough). Don't treat the epoxy fills any differently to the surrounding wood.
A little on the repairability of oil finishes
This is the theory the textbooks all tell us, that you can repair oil finishes and this is one of their strengths.
In practice I've had to repair oil-finished wood numerous times, sometimes during the finishing process3 and sometimes late damage to a piece long after the oil has fully cured. Regardless of whether the oil was fresh or cured, after only the first coat of fresh oil was applied the previously bare areas become virtually undetectable. After one or two further applications I couldn't find the damaged areas even knowing where to look.
1 In tricky situations (e.g. a flake taken off an edge) you can aid this process by providing "locks" for the epoxy. This could be as basic as scraping or carving in a small undercut or two, but many people just drill a few small holes at various angles. Obviously epoxy bonds very well to bare wood anyway, and normally you can rely on this bond to do all the work, but if you want the extra security it only takes a moment or two to do something along these lines.
2 With 5-minute epoxy despite it getting 'hard' in just 20-30 minutes I wait at least a few hours and often longer. With 30-minute epoxy, I'd wait 5-6 hours at least, maybe overnight. Using any epoxy with a slower setting time it's advisable to wait a minimum of 24 hours. Waiting longer never hurts, so if you get distracted or called away and a couple of days pass that's not a bad thing. Also remember that hardening and curing time is temperature dependent, so the cooler it is the more you want to err on the side of caution.
3 I've over-enthusiastically sanded through to bare wood near an edge or at a corner multiple times. Occasionally had to do a fill after the oil had already been applied just like you're asking about, and then had to plane/file/scrape/sand the fill flush which invariably removes all of the finish from the surrounding wood.