It's not the coils you have to worry about, it's the drivers.
Your horizontal rate is 15.x kHz and you want to go 30X
as fast, and you want to do it after a trigger event when
the CRT is free-running with a sync picked off some source.
Lock range I doubt runs 30:1. To sweep the horizontal 30X
as fast and keeping those coils, you will need 30X the drive
voltage if inductance sets the current ramp rate (and this
is not a bad bet - CRTs are optimized for cheapness in
manufacturing).
You also need a live vertical channel where the CRT control
is rasterized.
Beware CRT damage when the beam is allowed to sit in one
place (like when you're working on the drive scheme). Try
and back off the intensity a lot, the intensity you need for
a vector display is much lower than a full screen rasterized
display.
What about the new generation of USB digital 'scopes? I
have seen them with bandwidths well exceeding what you
say you want, and at a few hundred dollars. Many such,
and lower-end but serviceable no-name / little name regular
'scopes on eBay. Might try sniping for a while and see if you
get lucky before you get good at converting CRTs.
Oh, and one hand in your pocket if the other is in the guts.