the cart or the horse?

If you keep the omegas, which are great, stay away from amplifiers with high dampening. A quality set amp is the ticket. 

+1 charles1dad

Tube integrated amp. Add in OFC power cords from Ebay and decent solid core speaker cables.

I do not know those speakers but folks seem to say they are good. That leaves the source and the integrated.  I have the Bluesound and it is very good for the money. So that leaves the amp.  

By all accounts Hegel has gorgeous sounding amps but they can be pricey.  Used can be a crap shoot.  I would get something nice like a NAD C368 which works seemlessly either the Bluesound Node.  Your total spend would be $1K and transformative.  The little integrated would be light years better than the AVR.  I went on this same journey last year.  You can also try a Cambridge or an Arcam some such.  If you buy from an online retailer they will give you a nice in home trial period. Easy Peasy.  

Another alternative is to try a PS Audio Sprout.  Supposed to be a perfect little amp for your exact situation

finakky, don’t forget speaker placement  

Another option if you keep the speakers is the Schiit Ragnarok amp with their dac-card. About $1800 and has very good sound and adjustable gain which should make them match your speakers. 

i should temper my prior thoughts about a hegel or a nuprime... i did not realize the op’s speakers are a more 'niche-y' high efficiency single driver type that may indeed benefit from sweeter, lower powered tube amps... this may well be the ticket if the op is seeking a richer, fuller more saturated sound with those speakers

as others have said system synergy is critical in getting great sound out of a modest budget and speakers with very specific characteristics as the omegas seem to be (i have not heard them but i have heard many of their ilk...)