You will get better performance with separates unquestionably. It is uncommon to find a great preamp with a built in phono stage. That being said, there are some great units that will perform well and have very good phono stages.
This is a pain because of the wait, but AVM PA30.3 has an awesome feature set and is going to be a strong value at $2895 or $2995. I already have preorders for this thing. Expectation is March April for delivery. It has been available for a while in Europe. Best part is this will have variable outputs for easy amp matching. I can't speak to the quality of the phono stage yet but I would expect it to be good based on the strength of AVMs cards in the PA5.2.
Check out the Bel Canto e.One Pre5 ($2295) and the Bel Canto e.One Phono ($1495). These two units will be on budget, take up the space of one component, but are in two separate Chassis.
You need to dig into whether the output will match with your Bryston but the Rogue RP5 is an awesome preamp with a good phono stage built in at $3500. I spent a lot of time listening to it with a Rogue Medusa amp and B&W speakers.
The NAD M12 is largely digital centric but has a decent phono stage in it. At $3899 in is on budget and has great functionality. I am not sure if I would take this over the others but it is an option worth exploring.
All four of these are safe, well-made brands with good product support. I am an AVM and NAD dealer, I am currently vetting Bel Canto and have no affiliation with Rogue.