Looking over your system, looks like you did a good job of choosing… looks well thought out. I also think you are correct the DAC probably has the biggest opportunity for improvement. I recommend looking at a Schiit Yggdrasil. This is in your price range and is comparable to DACs at higher prices.
Spend time working with speaker placement and room treatments. They can be less expensive by using carpets or real purpose built acoustic treatments.
If you added a Yggdrasil, then all your components would be performing at roughly the same level and the next move, should you want to would be raising the whole system to the next level which would be 1. Speakers, 2. Integrated amp, 3. Streamer, 4. Power conditioner. My rule of thumb for this “level uping” is each new component should be no less that 2x an investment 3x is better. While this would could be one component at a time. The resultant system would be very substantially better.