Hi Ed,
First of all, how much trouble do you want to go to?
If you're willing to go to a fair bit, it might be worth checking the inherent runout of the ER40 chuck first, by putting a suitable ground bar in the 4-jaw and indicating it dead on centre, then clamping the ER chuck with a right-sized collet back to front onto the bar. This'll let you apply your DTI to the back of the chuck and check for eccentricity, axial runout etc. before you commit to anything!
Assuming all's good with the chuck, the next check is the D1-4 backplate - assuming you haven't set up a camlock backplate before, what you need to do is adjust the depth of the camlock studs to get the index line on the cams between the two arrow marks when locked - if they aren't it's not locked up and can loosen in use, eek!
Once the cams are right, check the spindle to backplate fit - I usually put three strips of Rizla Silver (the lightest weight) cigarette paper on the spindle face beteen the cam sockets and clamp up, if the paper is held rather than pulls out in all three places, it's fitted correctly. If the paper pulls loose, the taper socket in the backplate's too tight - unfortunately common with Chinese D1 hardware

This *can* be corrected with careful use of emery, a toolpost grinder would be better! It'll need to be held "bass ackwards" in a 4-jaw and indicated *dead true* bith radially and axially... IF the fag paper is gripped, remove and refit the chuck, and without locking it, the paper should pull free - this confirms that the tapers are mating correctly *before* the camlock studs pull it into an interference fit and against the face.
Once you're happy with the fit, stamp a "1" by the "1" mark on the spindle nose so you can reposition it when it's been removed.
OK, if all's good so far, onto fitting the chuck to the backplate!
Measure the recess in the back of the chuck accurately, vernier caliper will do fine, and record the size, do this twice or more ;)
mount the backplate and lock up tight, and turn the central spigot down to just (1mm / 0.040") larger than the chuck recess, then take a couple of facing cuts outwards from the spigot, finally taking a slight undercut around the spigot while turning it to size - this will ensure that any radius on the tool doesn't leave a radius that interferes with the edge of the chuck recess.
Try the fit - it should be a light (e.g. soft-faced mallet) fit, and should come apart with a wedge and mallet between the parts for the next step...
Mark through the fixing holes with a transfer punch, knock apart, drill and tap, reassemble and Robert is your parent's sibling!
Dave H. (the other one)