BAT amp + cheap pream vs BAT integrated vs Prima Luna integrated

You will keep those speakers, soooooo

1. Solve enough SS power for those speakers so you can enjoy them. New, or Used Amp, local to verify/listen with a warranty/return option.

1b. Select an amp that can be Stereo OR Mono, to have the option of a pair of Mono Blocks, get a 2nd one IF even more power is needed. Note: often, when bridged Mono, low impedance is not advised, or even possible. IOW, you find a great offer on a mid powered amp you think MIGHT be enough power, but only living with it for a while will reveal the truth, at least there is a way to get more power.

2a. Phono Stage, I am a fan of Tube MM and add a

2b. SUT for existing or future LOMC.

3a. Tube Preamp, with remote volume, or

3b. Vintage Tube Preamp, full featured, get remote volume from 

3c. Chase Remote Line Controller RLC-1

https://www.ebay.com/itm/397503797979?mkrid=711-53200-19255-0&campid=5338381866&toolid=10001&customid=51575700-090e-11f1-bc94-396434356230

My efficient speakers, I risked USED Tube Amp (Cayin A88T) and USED Vintage Tube Preamp (McIntosh mx110z with Tube MM Phono). I use a Chase RLC-1 for remote volume/mute/balance.

I enjoyed the mx110z for a few years, then had Audio Classics overhaul it.

When I finally got LOMC, I added a Fidelity Research FRT-4 SUT so I could continue to use the mx110z’s Tube MM Phono

IOW, If you get a nice tube preamp with tube MM Phono, and use a SUT for any Low Output cartridge, then you only need get a SS Amp and  A Tube Preamp with MM Phono to get going,

add a SUT later;

overhaul a working vintage item later.

Fully agree with @oddiofyl . The preamp is key.

If you cannot invest in two good separates go with a good integrated. Synergy with the speakers is also a key consideration.

Have you checked the new Simaudio all in one 371? One on the best value on the market right now for the money. Hegel also another one to check.

@tykozen - it’s difficult to suggest options as your choices tube vs SS AND uncertainty in future speakers are flux.

Speaker choices, especially sensitivity/efficiency, generally dictates acceptable amplifier power requirements.  A inefficient speaker will require more power. A low power amp requires an efficient enough speaker. So suggestions for your ATC may not be ideal for your next speakers.

Tube vs SS - you also seem unsure. Tubes typically have a warmer midrange, and typically have more limited power than SS unless you have larger amps with more heat and tubes. Tubes also allow tube rolling to customize the sound. SS typically performs better at the frequency extremes.

Integrated vs separates - you also seem unsure.  
Integrated (preamp+amp, sometimes include or optional phono stage, DAC, or streamer) will save you $ on separate casework and cabling.
Separates - offer more flexibility in future upgrading individual components. They also sonically perform better keeping the sensitive small signals in the preamp section separate from the high powered amp section. This is usually significantly more expensive than integrated with additional casework and cabling.

It seems the best choice is for you to demo speakers and gear to determine:

• tube or SS preferences 
• your next speaker - so that we can make better electronics suggestions 

I think there is a lot of gear out there that will make you happy.  You have narrowed to two items.  I have no opinion on that PrimaLuna. I have not heard it and have only heard older models. 
 

that said with BAT, pairing with an inexpensive preamp is a terrible idea. In fact, with BAT, assume you almost have to pair it with BAT to get the most out of it.  I am not a BAT dealer but have had multiple items in on trade including an amp, preamp, phono combo and testing each piece individually I thought it was awful.  Out of curiosity I set them up together and made a huge difference.  I know it will work with other brands but that hear has to have correct electrical properties AND be fully differentially balanced.  
 

I am not saying get primaluna, just don’t get a BAT and off brand pre.  
 

You have told us about your speakers and a little about your likes and dislikes but I am not understanding your listening habits, room size, types of music you listen too, etc…. Also, what kind of phono stage.  MM and MC or just MM.   something like an Audia Flight FLS 9 w/ phono might be perfect for you but need to know more about music taste, etc…
 

one thing that does seem clear is you do not like a cold sound.  Steer clear of SoulNote as it will likely not be to your taste.  It is wonderful gear, but will be a bit colder.