My burn rate chart shows Bullseye to be pretty quick burning. It's right below Clay Dot. But you're saying it's too slow burning to do what I'm wanting to do?
If you look at the load data for an extensive number of cartridges (I have), you'll see that Bullseye actually loads closer to Unique than to Red Dot/Clays. Not by weight, but by max pressure and the resulting MV. Bullseye is probably "faster" on the charts because of the high nitroglycerine content which will make the initial ignition "faster". When the nitrocellulose goes off, the rest goes off like Unique. That's why the old 45ACP military load was 5 grains Bullseye under a 230g FMJ RN. You'll never get Clays to get anywhere near that velocity without going over pressure. You can duplicate the load with about 6 grains Unique.
I've found that Bullseye can get close to Unique max velocities with about 10% less powder. The only downsides of Bullseye is the relatively high residual soot (even with full pressure loads and complete combustion with no/few leftover flakes) and the high nitroglycerine content tends to discolor and eat away at some powder measure hoppers. My LnL AP hopper is brown and mottled after 6 pounds of Bullseye even when loading in 1-2 hour sessions and emptying the hopper after each session.
I'm not saying don't try Bullseye nor will it prove unsuccessful as a light load, I'm just saying don't be surprised if you wind up with sooty cases, a very sooty gun and "spare powder" coming out of your gun if you use it. I did that extensively with 4.0g Unique when I first started loading 45ACP. With that gun, you also got you hair blown back with each lightweight shot. It just meant cleaning after every shooting session rather than once a month or so.