[jeffh]

Hi,
Thought I'd chime in as I've had some experience with what you're proposing. 'while I don't have the specs on what kind of halyard you're constructing (ie main, headsail or spinnaker), the 1/4" amsteel will after all be the "laboring oar" that is to say it will be carrying the entire load of the application. In this regard (again assuming) the double braid will be "integrated" into the eye/shackle for chafe uv protection.
I've found that on the halyards I've stripped the cover off of (main, headsail and spinnaker) that chafe protection is usually unnecessary. Primarily I've found this to be the case as the core (as opposed to the core and cover) is at such a reduced diameter that it actual chafes less than the core and cover.
The preceeding assumes that the sheave is the proper one for the type of cordage you've been employing. In this regard many of the applications I've done had originally employed wire to rope halyards, with the sheaves being configured with a sharper "v" shape to keep the wire from jumping. These type of sheave configurations will tend to chew up a naked core and would require some sort of chafe protection in the section of. Having said all that if the sheave(s) that you're looking at now are designed for wire rope they should be changed out for cordage anyway.
I believe that if you're going to attach some double braid cover that 7/16" may be a little too large for a 1/4" core, a better fit would be 3/8" LS, STA-Set braid etc.
For the stripped cover exposure to UV (while the boat is moored) I've just taken the cover that I stripped off the core and when securing the halyards after sailing, attached the old cover to the halyard shackle and hoisted the same to the exit box that way all the stripped portion of the halyard remains inside the mast and out of the sun.
The way I've found that works the best for what you're trying to do is simply make a locked brummel (I like Brion, have never forgotten to insert a closed bale shackle while forming an eye ha ha)
with the shackle inserted (if it's an open screw pin type you can insert it as you're drawing up the locked brummel) bury a length of at least 48 tapered diameters (1 ft. for 1/4" line) and you should be good to go without the integration of any cover material.
What your proposing seems to be a core to core splice (covered) after you've already stripped of the cover which seems a little like doubling the work you've got to do to achieve the same end result.
I'll quit rambling.
all the best