I don't have a clue how good the internal DAC in your Rotel A12 is, but I started my journey with a MacBook Air, an entry-level budget DAC, a preamp, an amp, and a pair of speakers. Today, I've moved on to dedicated streamers and mid-tier DACs. I can tell you the difference in every aspect of sound quality is night and day.
If you plan to stick with computer audio as described, do yourself a favor and invest in two things. First, get a good-quality DDC that provides excellent electrical isolation on its input ports and effectively attenuates jitter from computer streaming or local music playback. Second, invest in a good DAC.
I've tried both of the DACs you mentioned. Compared with older delta-sigma DACs, modern R-2R DACs such as the FiiO K11—and especially the more refined K13—offer noticeably better soundstage and transparency. However, I found their softer low-end response and lack of dynamics to hold them back overall.
Instead, I'd suggest investing in a higher-quality DAC in the $1,000 range, such as the LAiV μDAC or the Holo Audio Cyan 2. If I were you, I'd stay with an R-2R DAC.
For a DDC, I'd look for a used Denafrips Iris or a Singxer SU-6. Both are excellent choices that won't break the bank.
Lastly, for the computer itself, I'd recommend a fanless MacBook. Stay away from computers or laptops with cooling fans, as they can generate significant electrical noise internally and produce audible fan noise that can be distracting during listening sessions. Honestly, with a fanless MacBook (if you don't already have one) and a quality DDC, you might as well invest in a dedicated streamer like the Rivo Plus or a used Innuos Zen Mk3 instead. The total investment would be about the same, but you'll end up with a cleaner, more purpose-built digital front end and a better overall listening experience.