A couple of issues occur to me.
First, an annular cutter is the way to go if you must save the core. However, that reminds me of a favored saying of college. If he thought one was on the wrong track, he would accuse them of looking up a dead dogs arse and would tell them "Stop that. There's nothing up there that you can use"
The point is, compare the cost of purchasing a 5" long x core diameter bar of the same material and compare that cost to any save-the-core option. If I had a 2.25 diameter x 2.5" deep annular cutter I (a hobby machinist) would probably charge you more to do the work than the core is worth. Of course, I'm assuming your "solid rod" is not gold or other precious metal and disregarding any "save the Earth" motivation. Recycle the swarf.
Trepanning (single cutter), in general, is a special skill and in the dimensions you mention, it's quite unreasonable, IMO.
Another option is to search out a 2.25 diameter hole saw that can cut 2.5" deep. Hole saws are commonly available as Bi-Metal construction with hardened teeth or even carbide teeth. I suspect such a hole saw (+arbor) could be found for $50-$60. Still probably more cost than the core is worth, but maybe you already have a suitable arbor (and a hole saw?). The problem with hole saws, in general is that you have to FREQUENTLY withdraw them from the material to flush and clear the cuttings. But, rig-up an attachment for your cordless drill to spin the hand wheel on your tailstock so you might be able to get the hole through while avoiding physical and/or psychological therapy. Keep the cut wet to keep the teeth cool. It sound like a mind numbing job, but that's why it's just a hobby.
You didn't mention the size/power/model of your lathe or how large a twist drill you have available. If you have sequence of twist drills up to 2.25" diameter, that would be the cheapest, quickest way to rough out the hole. If you don't have the big twist drills or your lathe doesn't have the power/rigidity needed to use them, then we're back at a mind numbing task of boring a lot of stock, but it can be done.
If you still need to save the core, wire EDM is the way to go. Not usually thought of as an inexpensive process, however the core will be saved, to the max, and the hole will be to finished diameter (no boring required). Do you have a friend with wire EDM?
My recommendation is to save that precious "solid rod" for a yet-to-be-defined future project and get a piece of aluminum,
like this, to work with. The sleeve will be used in compression, so aluminum will be perfectly satisfactory and much easier to machine (I assume).
Regardless of the sleeve material, the much smaller over arm (2.375" vs 3.5") may result in a significant reduction in the rigidity of the M head mounting. My concern is hypothetical because I don't know how cantilevered the M head will be on the Van Norman. If the finish produced by the M head turns out to be unsatisfactory, take a look at adding some additional struts or ??? to stiffen the M head mounting.
If your lathe is up to the work, get a piece of aluminum and make chips.
You're right next to Baltimore? It should be pretty easy to find a piece of aluminum. Go to a few local machine shops and ask if they have a drop that they can part with. You should now know how much it's worth to you. Ask if they have wire EDM and how much $$$ (Ouch!)?
