You do not need a centerdrill.
Centerdrills are for making centers in a shaft in order to use a live or dead center, but it's "convenient" for some to use for drilling, but sometimes even centerdrills are off . Also they are 60 degrees and not 118 like a drill.
Do what machinists have done for 100 years, before they had easily obtained centerdrills.
Process
When ever you want to drill in the lathe, always place the bit in the chuck with the cutting lips/edge of the drill in a horizontal position. Bring it up to the work face ...then using your tool bit and cross-slide ,( raise the tool bit up about 1/8 ") and then move it in maybe 1/4" to 1/2 " behind the cutting flutes and push the drill off center about .010" ( the above depends on the size of the drill-) . Now this is a two hand operation advance the drill so the divot in the work piece is about 1/2 the drill size and when that point is reached -keep feeding the drill in and at the same moment retract the crossfeed !
Now you have a perfect centered hole start !
Technical Data explanation
So here is what is happening . Pushing the drill away from you makes it cut only on the far side cutting lip( it becomes like a boring bar in miniature) The result is a conical shaped hole with a *** in the middle and is a true centered machined surface ( like a boring tool). Now what you want is a cone length longer than the *** length - when this is reached ( 50 % to 90 % diameter) and the drill forced in (without side cross-feed pressure !) the hole forces the drill lip inward as the force needed to cut the *** is less than allowing the drill lip to cut the cone .
A few notes
Doing the above saves setup time . The machinist immediately goes to his drill , no swapping chucks or using the Jacobs key.
With Aloris Tool Posts , it easy to raise the tool bit and then drop it back into place as I suggested here, but when the old guys used Lantern Posts, they just made sure the tool bit's cutting edge did not touch the cutting flute of the drill bit and used the "Heel" of the toolbit when touching the drill bit.
In fact, the old guys did not even "face the end" of the work before drilling, and used the above method to "mini-face" the saw cut stock with the drill a small amount before proceeding as noted.
All the above is for those who wish to expand their skills .
It also adds to satisfaction at using your skills in new ( or old ! ) ways
Rich