The DI should have no impact on the sound that comes from your rig. Any degradation you are getting should be coming from other sound sources that are using your signal.
I generally do not like having my bass added to my monitor mix. Frequently this creates the sensation that the monitors are fighting with my rig, rather than working with it. In actuality, they sort of are fighting because the problem is destructive interference called comb filtering. Comb filter occurs when you hear the same sound coming from two or more different sources, and the arrival time of the sound from each source is different.
Part of the problem is the different distances from your ears to each cab, and part of may be the difference in how your bass cab and the monitors are tuned. If you used identical cabs for your rig and the monitors, you would get the best sound by spacing them so they are equidistant from you head. The idea is so the sound from each source arrives at your ears at exactly the same time. When the spacing is not equal, the sound from each cab arrives at your ears at a different time which results in phase variances between the two signals. At frequencies where the signals arrive in phase they add, and at the frequencies where the signals arrive out of phase they cancel.
When you have reflex designs with significantly different tunings, you have another type of phase variance. Each tuning will have somewhat unique phase shift and group delay characteristics in the frequency band where the port is active. So even if the different cabs are equidistant from your ears, they may fight around the port tuning frequency. This can create a perception that the sound is hollow. or lacks power and focus.
I recommend some degree of compartmentalization. Worry primarily about your stage sound, and avoid trying to maintain 100% control over how you sound out in the house.
The reason, IMHO your sound needs to be processed for intended use and location. The sound you desire on stage will change as it travels out into the audience. In other words, the great sound you get on stage will not sound the same out in the audience. If your amp is pushing the room, it's fairly typical for the group leader to go out in the house during sound check to call tone changes back to you on stage. The idea is some degree of priority is given to making you sound good to the audience, and this process can degrade the quality of sound you are getting on stage.
Also, the tone and amount of compression you find ideal on stage for your monitoring purpose may not work well in the house mix. In order to do the job, the audio tech requires some degree of artistic freedom to shape all of the sound sources in a way that allows him/her to build a good sounding mix. Ideally the audio tech should have a clear idea of what sort of sound you are going for. The end result should be in the same ball park, but it's really not reasonable to expect it to be exactly the same as the sound you hear on stage.
The reality is it can be difficult if not impossible to deliver a really good bass sound to the audience. In my experience the quality of the bass tends to vary significantly from one gig site to the next. This is just the nature of the beast. Having great gear does improve the quality, but cannot eliminate all of the problems that degrade your sound. To some degree the most important thing is to recognize the limits of what can be achieved and adjust your expectations accordingly. Focus less on having perfect sound and more on making perfect music. IMHO great musicians anticipate this problem and work through it.
On rare occasion you will experience a situation where are all of the factors that degrade your sound seem to work out by serendipity. When this occurs, enjoy it because it will not happen frequently. Try not to draw any premature conclusions from the experience. For example, if you are playing a backline rig that it new to you, don't assume the rig is solely responsible for the great sound. It's likely that if you use the same rig at a different performance venue that the sound will be totally different. I.E. the rig may have little if anything to do with having great sound on a given gig.
Sorry I know this is probably not what you wanted. But it is, unfortunately how things work IMHO.
My approach as a bassist: My preference is more towards full-range, flat-response. However, long ago I gave up on the idea of having a really fat bass sound on stage. I find the clarity and focus is the most important thing I am looking for, and cranking up the lows beyond a certain level tends to cause masking that degrades the clarity and focus in the mids and highs.
My approach is typically to elevate my cab and tilt it back to aim at my head. The dispersion of a bass cab will change with the frequency. At low frequencies the cab is essentially omnidirectional. As the frequency goes up, the cab will start to force more and more of the sound out front. At higher frequencies the sound from one cone drive will form into a cone shaped beam. If the cab is flat on the floor, this beam of sound will shout out way below your ears, so you won't hear it.
Elevating the cab reduces the lows and puts a notch in the low mids related to the height of the cab. Tilting the cab back to aim at your head ensures you hear the full range of the direct sound coming from the cab. If you prefer clear articulate sound, I suggest trying this approach. However, if you like big fat lows, you probably won't like it.
