Well, tough question. I do not know the answer, but I can tell you some of my findings related to TT2, to M Scaler and also to Elise when combined with them.
1. Regarding influence of M Scaler in my setup. For me, using only headphones and TT2 in DAC mode (I do not use the integrated headphone amplifier of TT2 - the AMP mode), when switching M Scaler from Passive mode to Full upscaling mode I hear only a marginal difference. It is there, you have a small idea of better definition and separation, but I would say around 10%. A bigger difference for me is to completely remove M Scaler from the circuit. Strange, I know. Might be related to my own setup, preferences and so on. It is hard to say. On M Scaler thread you see hundreds of pages of debates, and it is hard to draw a conclusion.
2. Regarding TT2 as a DAC compared with Qutest. When I upgraded from Qutest to TT2 I had a feeling of better resolution, bigger sound stage but also a warmer sound. I kept both everytime on Incissive Neutral filter and listened through Elise. This means that adding Elise to TT2 as an external amplifier might be too much if you plan to have a neutral sound. It depends on which headphones you pair with. There is a big difference from what you hear using T1.2 compared with HD800 or ADX5000. And it is also a matter of personal preference. I like to have that warm analogue sound from the 70s when listening to Led Zeppelin or Black Sabbath. But this kind of sound might not be perfect for classical music or orchestral. However, Elise is like a chameleon and exchanging tubes might transform enough the sound to match your preferences, in detriment of your wallet. And TT2 needs also a linear power supply to shine. And if you add M Scaler you definitely need better dual BNC cables besides the supply. Both made big difference in my setup and you should not ignore them. And for all of these I recommend to use a power conditioner. I had benefits also for Elise, not only for TT2 and M Scaler when connecting my whole headphone line to an AudioPlan FineFilter S and TT2 + M Scaler in addition to a Furman AC210A-E.
3. Regarding how TT2 integrated headphone amplifier sounds compared to Elise. Well, I was never too good in describing sound using fancy words, but one thing you hear first when you try TT2 in AMP mode are the clarity and incredible dynamics. I had a feeling that everything happens faster, everything sounds crystal clear. But when I tried it with my 600 Ohm Beyers I also had a feeling that the bass is under my expectation as quantity, that the sound is relatively thin. Everything was technical, clean, precise but my T1.2 sounded much better on Elise. Again, a matter of preference. On the TT2 thread many praise it with Empyrean which are a dark pair of headphones with low impedance. Most probably those compensate on what I felt it was missing with my headphones. I cannot say since I never listened to Empyreans. On Elise everything happens a little bit slower, but each instrument weights harder. And a big advantage of Elise is that it is very responsible to tubes, it has countless possibilities of transforming the sound. On TT2 you just hear it, that's it, you like it or not, you have to get used to it. Of course, you can exchange a headphone cable or use some equalizer but I think the change is minimal compared to what you can do with Elise.
And now I'll try to answer you... If I will have at the moment TT2 on the table the first thing I would do is to obtain an LPS for it and compare how it sounds after I add that one. Then I will plug TT2's LPS into a power conditioner. Only next step for me would be to add the tube amplifier, and compare both external and internal amplifiers for some time. Thinking that both these upgrades (LPS+filtering and tube amplifier) are around half of M Scaler's price and that they will provide you more room to try and experiment compared to adding the M Scaler, which you just plug there and that's it, I think they are the best choice first. Of course, if you can try the M Scaler with the possibility to return it, you can do it right away. This is how I see it, let's hope others will come with more opinions. And of course, my answer might be related to my own setup and preferences as I already said. Adding power filtering and the LPS will, most probably, remove some brightness and harshness from the sound, and only after that you will know if you need also a tube amplifier or not.
Well, I am sure I have not properly provided an answer but let's hope my findings help.