My system is dual use but very high end. The solution is to use the HDMI direct to your video. Forget surround, that's where all the bad sound is coming from in the first place, and feed just the L/R to your integrated. Exactly what I do. You won't have surround but you won't miss it because the money you save on amps, cables, and surround speakers buys you stereo sound that more than compensates. Especially when that is what you care about most anyway.
My advice, keep what you have and with $3k of your $4k budget add a Swarm/Distributed Bass Array type subwoofer system. This will be a huge benefit to both music and movies. Then with the remaining $1k upgrade your weakest wire and if anything is left after that then some Blue Quantum Fuses or HFT, depending on how much is left. Together these will transform your system in the direction of "you are there" palpable imaging and immersive, articulate, powerful and deep bass like you never heard before.