What adapter (bullets) & Hardware must I buy to connect LDF4-50 Heliax to this BIRD?

At HF frequencies, and ham power levels, not much. A small impedance bump if the diameters aren't matched exactly. Sooner or later you have a bump anyways when you transition from the inner conductor down to a connector pin. A solid rod would work best, machine one end to fit the inner ID, machine the rod OD to match the inner OD, and drill a hole at the other end for the connector pin. The outer conductor will depend on what's available in the local plumbing shop, I don't think you want to buy a solid copper or brass rod almost 2" diameter in order to machine 99% of it away.

The 1-5/8" line capability far exceeds what the load is rate for (20 KW at 50 MHz), so I don't think you have to worry about arcing!

Don't overthink the RF side, treat it like plumbing and get the dimensions as close as possible and it will work fine. We used to make matching transformers for 75 Ohm cable TV hardline out of hobby brass tubing and hose clamps, and used them from 144 up to 1296 MHz.

 

If you want more info on rigid coax, or want to see some drawings, check out:
http://www.myat.com/technical
http://www.dielectric.com/inc/catalogs/Transmission_Line_Catalog.pdf