I take it that you are masthead rigged, that the forestay and the foresail go all the way to the top of the mast.
Right, masthead rig. I have two masthead halyards at the front of the mast--jib and spinnaker, and the main halyard at the back of the mast. The headstay connects at an elevation between the jib and spinnaker halyards (masthead pic from
@bgary )
I have a question: I think from the post that you climb the mast using 3 halyards. Is that correct?
I often climb from two halyards but I've never climbed from three. I just made up my own rule that my third safety rope would always be a dedicated line that is secured directly to the mast. While I was doing mid-mast work, my 3rd line was one of two eye-sliced 5/8" dock lines suspended from the spreader bases.

So, yeah, as I transited up from one safety line to the next, I'd unclip from the lower safety-line prusik (leaving it in place for use on my way back down) and wrap a new prusik around the upper safety to clip myself onto. Of course, when changing prusiks connections aloft, the basic rules are:
1. each prusik is clipped to your harness with its
own screw-gate/locking carabiner
2. never open the gate of a loaded carabiner, and
3. always have your primary and first-backup prusiks supporting your full weight before opening the gate & swapping prusiks on your 3rd carabiner.
Once I started doing more work at the masthead, I hung dedicated climbing lines from carabiners attached to the masthead apparatus itself.
When you start replacing halyard sheaves, you have to be flexible about your climbing lines. The sheave you are replacing has to have a slackened halyard. When I removed the main halyard sheave on my last climb, I ascended on the lines shown in the photo above, but then had to remove both of those carabiners/lines from the masthead to get access to the main sheave itself.
A fractional rig would complicate the issue. I think you said you tried a prusik hitch around the mast itself with good results. I have done the same with a webbing loop and a klemheist hitch (post #25, above). I think that makes a suitable safety until you can top-hang some lines. If you're going to spend much time at the top of your mast though, I'd say you need to find something secure you can hang dedicated climbing lines from.