Yes, you can use Sunday Keys patches in a DAW, and the best one to use is Logic Pro.
The reason is that Logic and MainStage are sister programs both made by Apple that both share the same plugins and effects. This allows you to actually import patches and channel strips directly into Logic and record the MIDI data there. This is very handy if you want to get into quantizing MIDI, editing the MIDI note score, editing MIDI velocity of a recording, editing MIDI automation (like filter, etc.) over time.
By far, using Logic to record MainStage patches will give you the most flexibility out of any DAW due to that shared patch/channel strip file structure. Note that Logic is a DAW, and so you don’t get the same visual GUI with all of the mappings like you do in MainStage, but you can still achieve all of the same sound effects in Logic that you can in MainStage.
You can save a channel strip or a patch from MainStage to use in Logic Pro, check out this article for more tips.
In any other DAW, you can record/work with MainStage raw audio.
You can do this by either recording directly in MainStage via MainStage’s record button and then transferring that audio file into the DAW of your choice. Or you could use a “virtual audio cable” program such as Sound-flower to directly route the MainStage audio into your DAW. Sound-flower will also give you the option to set different virtual output’s per channel strip- so you could set things up to multitrack the audio recording of MainStage into a program like Reaper.