Hi Celo,
The C7ES3 and SHL5s are my favourite speakers in the Harbeth line and I could live with either one. You are right, the SHL5 Plus has a bit of the "hi-fi" sound as it’s currently the most lean sounding Harbeth, followed by the warmer C7ES3 and the SHL5 which is even warmer.
Yes, I still have the Naim with the SHL5s. In my experience, you have to spend a lot to get a considerably good sound from the SHL5s if staying with Naim. For this reason, I cannot recommend the Nait 5si for the SHL5. I have used Nait XS, 202/Dual Teddycap/200 and now 282/HCDR/250DR with full loom Chord cabling throughout. The latter is a massive improvement over the other two Naim systems in areas of transparency and openness. In my experience the amplifier forms a crucial link with the Harbeth. With the wrong amp, the Harbeth will sound overly warm, lacklustre and dull. With the right amp, the speakers will come to life and show superb dynamics and transient attack with great articulation and control. The sound will just pour out freely from the speakers with a lot of poise - reduced smearing of notes and improved articulation and refinement. The energy and tonal colour of instruments will sound different even by fiddling around with the cabling in my Naim system.
In summary, I would advise against the Nait 5SI. I do not have experience with this amp but my experience with the higher level Naim tells me that the SHL5 will not show its full potential with Nait 5SI. One integrated amp you can consider is the Sonneteer Orton. Croft pre/power are also reported to be a good match to the Harbeth. The higher range Naim is good stuff but it gets too costly when one is enticed by all the upgrade options. It may be a better choice to stay away from Naim to avoid getting into the upgraditis syndrome.
Good luck.