I have a feeling that if you choose wisely with the turntable and the speakers, the integrated amplifier might prove to be the component you upgrade next. If that’s correct then you could skimp a tiny bit on the integrated amp. This doesn’t mean choose something that isn’t any good. It just means spend less money on it.
Something very affordable comes to mind. At one point in time I had an integrated amplifier in my office made by peachtree. It was not expensive, sounded very nice, and even had a DAC built in for a digital source. The model I had was the deco. I believe you can find the nova by peach tree for very good prices either refurbished or used. There is a peach tree deco on audiogon now for $500, which is a great fit for your budget.
$2200 for your table, you already have a phonostage, $500 for the Peachtree, that leaves $2300 For speakers and cables. Speakers are super personal, but I would put these brands on your radar: Elac, Totem, Magnepan. I would spend as much money as you can and speakers! Elac is making waves in the audio press for offering high end performance for a lower price point. $2k on an Elac speaker may be impressive.
As for speaker cables and interconnect cables, I would recommend going with The entry level from a well respected manufacturer such as Kimber or Cardas. You can get a pair of Interconnect worth it’s salt for $100, and short pair of decent speaker cables for $200-$300. Check out Kimber PBJ and Kimber 8TC
Good luck!