Hello and welcome ! :)
Now that's my cup of tea ; lovely sound and playing. Proof positive of why the Telecaster is the solid bodied guitar of Jazzers.
As one who owns a few of the clearer, vintage B.K.P. ; my observations are as follows :
I seem to be hearing a more pronounced mid range and a touch more thickness on the top in that sound, than Alnico AIV or even Alnico II Mondays have sounded in my Archtop, or Les Paul. They did get more like that in my all Mahogany S.G. though. The Stormy Mondays are very nice, open - and sweet ; but the E.Q of them is relatively flat / even across the range.
The Missisippi queens can sound simultaneously wonderfully 'chewy' and clear - but may not yield quite as much fine detail when fingerpicking.
The Manhattan ( Not on the B.K.P product page, but are made upon request ) is my favourite B.K.P for thicker,woody, but detailed sweetness. They are B.K.P's take on a 1950s Jazz 'P90' type pickup, in a humbucker size.
In any case, it will depend greatly on what woods, and what amplifier you are using them with.
My assumption would be a milder 'Mule' - having a lovely subtle mid range hump. The AIV type Mules are in my Les Paul, and will remain there ! By 'milder' - I am thinking of using an AIII magnet , instead of the traditional Alnico IV. The Manhattans use AIII magnets, and they are magic at balancing just enough thickness, without getting 'pushy' .
As always, remind us of your guitar woods and rig e.t.c. ( especially if our resident Luthiers are viewing ! ) . If in doubt, contact Tim at B.K.P .
Although I have placed the pickups I am about to re-iterate in a pair of Hollow / Archtop Jazzers - you may want to check out my sound clips for the Manhattan , and subsequently both the Alnico II and Alnico IV Stormy Mondays in the 'Players' section. I'm not advertising my clips :lol: - but you will at least hear the difference between the 3 most obvious candidates.
I have no direct experience of an AIII Mule , but can imagine the character my AIV's would assume with a softer magnet. As always, I hope as many people as possible 'chip in' . Just to re-iterate - if in doubt, Tim Is the man !