Jason,
Have you got a dial indicators with a finger? if so, you can get, or make, a clamp to hold it in the spindle and then sweep it round by hand and when (as you tweek X/Y) there is no oscillation on the indicator, then you're centred (this is also how a Blake or co-axial centring indicator works).
Another approach (which really needs DROs or negligible backlash) - take a contact edge-finder type probe and creep up in, say X at approximately the widest point; when you make contact, note the X reading; then wind X until you make contact on the other side, note that reading (or the number of turns etc. it took to get there) and halve it - that will be your central X value; then repeat for Y.
Dave