Hey Charona, did you also connect one or more audio outputs from the DBI to the input(s) of the UMC404HD?
This way you should be able to arm tracks and record both midi as audio signals into Live.
The UMC is a very capable device and there is no need to use the USB other than for the MMC.
I guess you have some audio monitors (or headphones) connected to the UMC. With the Mix knob you can listen to a mix of the direct sound of the DBI and the recorded and/or effects processed sound from Live.
It depends on the speed of your computer (and background tasks), how much latency there is between the input and output signal.
When you connect the midi-out from the UMC to the midi-in of the DBI, you can have Live playback the midi notes on the DBI. To prevent double ghost notes, you need to turn “local off” as explained in the manual (9.10.2.15. Local Control).
I hope my explanation makes sense. Please let me know if you can record the way you want.
For I know audio and midi routing can get confusing to setup.
[edit: add USB and latency remark]