First, I’d say “try it with the DAC you have”, before you do anything else.
Beyond that …
Most of the reviews for Bifrost MB I have seen were done on what I would call “original” versions of the DAC. Those sound rather different, and not as good, as the units I’ve heard that were manufactured starting in October 2018 (it’s possible whatever changes are at play go back before this, even, but that was the oldest unit I had heard that exhibited them).
“Original” Bifrost MB was almost impossible to distinguish from Modi MB. So, new USB Gen 5 interface not-withstanding (which was a further improvement over the old one and the one in Modi MB), there wasn’t a lot of reason to buy a Bifrost MB unless you wanted upgradability or an aesthetic and size match with the other amps in the same sized chassis.l
“Current” (i.e. units newer than ~10/2018) Bifrost MB sounds notably BETTER than both to the “original” one and to Modi MB. It’s closer to the original, pre “A2” (since it’s not an official thing) Gungnir MB than it is to the old Bifrost MB or Modi MB.
Still … it’s $600 so outside your budget, but I wanted to clear up the differences. I’ll have more detail on that in a-soon-to-post review.
Regarding the Airist R-2R DAC (RDAC) vs. these DACs, I would rank it like this:
Bifrost MB (Current) > RDAC > Bifrost MB (Original) >= Modi MB
RDAC with Lyr 3 is a decent pairing. There’s no particular “magic” or “special” synergy there. But there’s no particular issue either. Though I would tend towards more linear/less “warm” tubes with that combination.
As for other DACs … there’s a good population of them around the $100 mark. The bulk around $350 are DAC/amps and in most cases not notably better, as DACs, than the $100 or so pure DACs, unless you needed balanced output (not relevant to Lyr 3).
The Pro-Ject Pre-Box S2 Digital is probably the best you’re likely to find, without paying for more stuff than you need to. The headphone output is anemic, but isn’t very involved in it’s implementation, so it’s not wasting much of the BoM on that. It’s not pointlessly doubling up on parts to give “balanced” output that you won’t use with the Lyr 3. It supports pretty much every input format/standard going from high-rate PCM and DSD to MQA, if you care about such things. Has a very competent USB input (and can be USB powered or externally powered if you want).
The Topping D50 is similar, in many regards, to the Pro-Ject Pre-Box S2 Digital, and $100 cheaper, but it has an annoying power arrangement, has weaker format support and it got fatiguing fairly quickly where as with the S2 I didn’t run into that.