Considering buying a CD transport

My current disc spinner is a Cambridge Audio 751BD which I'm now mostly (except for SACDs) using as a transport, as I vastly prefer the sound of my Bryston BDA1 DAC to what comes through the CA. Even through the Bryston DAC, CDs can sound shrill. The CA works extremely well for DVDs, Blu-ray discs and SACDs, but I don't think I'm enjoying my rather extensive CD collection as I should. Would a separate transport make enough difference to be worth considering? I've been reading reviews of the Cyrus CDt which are tempting, but I'd like to hear some real-world opinions first. Unfortunately, there are budgetary constraints, so I would like not to exceed about $2K.

The rest of the system: Hegel H200 integrated, Proac Response D2s, REL S2 sub, Clarus Aqua analog interconnects & speaker cables, Nordost coax.
I'm intrigued by the Pro-ject CD Box RS, but have not auditioned it. Well under your budget.
I also have a BDA1 which I use with a tube pre and an Arcam DV-137 as transport.. The shrillness could be the DAC/CA combination. I do have some CDs that sound shrill however mostly only early rock ones with analog to digital conversion. Any way you could try out another CD only spinner with the BDA1? A marantz 5004 or 5005 player may make for a better transport at not a great cost ($350).