I think it depends on the guitar.
If an overwound A5 PAF like the Birmingham sounded shrill in a guitar, the Slash is likely going to be very similar. Birmingham is an 8.7k pickup with 42ga wire; Slash is an 8.9k pickup with 42ga wire. I haven't found that A5 vs A2 is going to be a massive shift in the high end, more subtle. It's also going to have a little less low end which could emphasize the shrill tone. And just from experience, the Slash bridge had a push in the upper mids, not harsh in my Les Paul, but definitely forward sounding.
@Stone If you aren't familiar, PAF's are wound usually between 7.5k - 8.5k resistance with 42 gauge wire. High output pickups like the Duncan JB and Dimarzio Tone Zone are wound around 16k - 17k resistance with 44 gauge wire. Medium output pickups like the Duncan Custom range or Rocksong are 12-13k resistance with 43 gauge wire.
That's where I think if you want bigger sonic changes, don't jump from PAF to PAF that are very subtle, but try a different type of pickup. Believe me, I've gone through the PAF carousel a bunch of times...
Also, you could swap the A5 magnet from the Birmingham for an A2 and you should be very close to the Slash pickup in terms of specs. Magnets are about $5 and it's pretty easy to swap if they don't have covers. Just loosen the screws on the bottom of the pickup and push the magnet out the side. Might have to soften the wax a little if it's cold. If they do have covers, you can cut the solder with a box cutter (fresh blade) easily and pop them off. Not a bad idea to know how to swap magnets. Just make sure to match the polarity so it's not out of phase.
Lastly, gotta put a plug for the lowly Dimarzio 36th pickups. I did the pickup carousel a bunch of times on my Les Paul, including the Air Norton, Wolfetones, Suhr Thornbuckers, Seth Lovers, Slash, Suhr Aldrich, Friedman Classics, Duncan A2 Pros, and probably a dozen others. Currently put the Dimarzio 36ths in, and they are really nicely balanced on this guitar with less harshness in the bridge and less mud in the neck. You can find them used cheap all day long.