I would use filled epoxy for this, it's probably the best thing going as it's very tough, bonds very firmly to the wood and can be extremely cheap. Possibly best of all you make as much as you need when you need it without any worries of buying more of a product than a single project needs and having the rest go off sitting on the shelf
Once fully set epoxy fills respond very well to working with sharp tools and in should sand excellently too (although it can end up very hard, much harder than the surrounding wood).
You can use any type of epoxy, from the cheap stuff in the dollar store to an expensive 'epoxy system' such as those sold by West Systems and others. They all seem to work about as well. When doing a large fill a longer setting time can be advantageous — especially in warm weather! — but even for a fill of this size I've used 5-minute epoxy and had plenty of time.
Speaking informally when this is recommended people usually just say to use sawdust, and literally sawing dust mixed with epoxy can actually work fine even though it can be relatively coarse. But if you want a very smooth finish and a more uniform colour use the finest dust you have available.
In boatbuilding where filled epoxy is used widely as a construction material it's not uncommon to buy commercial wood flour for this purpose, but sanding dusts from an ROS are generally fine and work about as well, especially if created from paper finer than 180. I've had pretty good results using the dust from a worn 80-grit belt so you don't have to be too fussy about this though!
Blend the epoxy together first, to ensure it is properly mixed, then mix in the dust.
You can mix in just enough dust to colour, the often-cited "peanut butter" texture (literally looks like peanut butter if the wood dust is the right colour) or much denser than this, a paste stiffer than spackle/patching plaster. They all work fine in the right context, what'll suit best is partly up to you and partly dependent on the nature of the void being filled.
For a through-hole like this, using a mix of any sort of liquid consistency you'll want to back the hole with tape to stop it simply running out the other side of the hole. Masking tape or painter's tape are usually used but plastic packing tape also works excellently.
Other fillers and colouring agents
To colour only:
- oil paint
- enamel paint
- dry pigment (glow-in-the-dark powders are currently quite popular)
- alcohol-based wood dyes
Be cautious about adding too much of any of these or you'll severely compromise the strength of the epoxy.
Tinted epoxy looks great but in general you'll get a stronger fill when you fill the epoxy with some type of dry dust. Nearly anything can work:
- sanding dust
- wood flour
- metal powders
- talc (also used as an additive, to improve sandability as it's very soft and lubricant)
- fumed silica (also used just as a thickener)
- plaster of Paris or other types of dry plaster
- crushed stones (turquoise is currently very popular)
- coffee grounds (my preference is used, and very thoroughly dried)
- wheat flour
Those last few are not jokes!
I happened to be reading this just yesterday, using epoxy and coffee grounds to fill voids on turned bowls, on Doc Green's Woodturning Site.
Note this from the above link:
Generally speaking, I wait at least two hours before I attempt to sand the filled area. A delay of four hours is better; and overnight is better still.
It should also be food safe.
This is a contentious issue in woodworking circles. What one person considers food-safe another will not, a great example being thinned varnish (some versions of "salad bowl finish" are just this) which some refuse to use even for the stated purpose, but which others are comfortable to use on a cutting board.
The nature of the food contact can be considered an important factor as well, what's OK for incidental food contact (such as you'd get on a kitchen table) is not necessarily the same as with a food-preparation surface where raw food will routinely be in contact with it.
As a general rule, any finish that dries/cured hard can be considered food safe once fully dried or cured. Any toxic ingredients (and there can be some, usually in minute quantities) are safely locked away in the matrix of the hardened finish. This includes heavy-metal driers in some wood finishes, e.g. BLO and varnishes.
Now what about epoxy? With wet starting products are most definitely not safe, at the least they are irritants and you do want to be careful to clean off any accidental contact when using it. But once properly mixed and fully cured epoxy is generally believed to be utterly safe, safe enough that it can be used to coat wooden drinking vessels.
So I think it's perfectly fine for something like this myself but your mileage may vary.