Electronic – What’s the purpose of the full wave diode bridge in this DC-DC power supply

You would have to disconnect the ground on the transformer center tap, but I would not recommend doing this because the insulation between primary and secondary (or between secondary and case/mounting) may not withstand 12 KV.

If you need 12 KV, you should get a proper 12KV power supply.

Well, at a current \$\frac{1}{\sqrt2}\$ lower your \$I^2R\$ heating will be half, so the overall copper heating would be the same as with the bridge.

Win Hill et al. are simplifying things a bit here by ignoring the diode drops. In the case of a requirement to get a very low voltage where the diode drops make significant difference, and/or if the granularity of the lamination size selection leads to a transformer that has more capability than required, the centre-tapped configuration may make a lot of sense.

Also, not all the transformer losses are copper losses, so doubling copper losses does not increase the total transformer losses by 2:1, especially with cheap lams.