I don't think I will be changing my Lyr amp now as I've just bought it for my new LCD-2 based on many positive opinions. I like the combo (LCD2+Lyr) and I think it's good enough for the money. Now I am looking for a suitable DAC for my headphones, one with large soundstage width. My budget is preferably $400-500 (max $700).
Bifrost is good in its tier of performance, but it doesn't sing like DACs in the $750-and-up range. Since you are using the LCD 2.2 (which is an expensive, highly resolving headphone), you will easily hear the difference. I wouldn't recommend it in this application. Instead, I'd consider upgrading the amp.
In terms of sound quality, a Decware Taboo amp would outdo a Lyr + Bifrost combination for roughly the same cost. The Taboo is built for low-impedance headphones like the LCD and has a tube crossfeed mode (called "Lucid Mode") which noticeably expands the soundstage. Furthermore, when and if you decide to add a dedicated DAC in the future, it will scale that much higher and bring the LCD to a whole new level. In other words, you wouldn't be "locked in" to a lower level of resolution at the source.