Nice setup! I've been thinking about getting a Sony VPL myself.
#1 How do you like it?
#2 What reason(s) made you go with the 675 over "just" the 365?
I'm an AV nerd, so buckle up :)
Don't get a 365, as it has pretty glaring limitations (briefly had one). It is limited by a max HDMI bandwidth of 10.2gbs, versus 13.5gbs in the newer models (285, 385, and 675).
Both the 285 and 385 are totally viable options. For the budget conscience, the 285 is a solid value at $5k. It doesn't have Lens memory or an iris. The 385 adds these features for $3k, which is steep for what they are. For most people, this translates to better black levels for movies, and basically nothing for games because game mode normally disables the iris.
There is little difference between the 675 and 385, other than 300 lumens. I only got a 675 because I was offered a great deal from a local company that took my old projector as a trade in towards a 675 they had in stock. If it wasn't for this deal, I would have purchased a 385.
After a ton of research and viewing, I narrowed my options to the Sony 385 and JVC RS520/540. It's a battle of native 4K (Sony) vs. better contrast (JVC). I'm not a fan of the Epson 5040 compared to these two, it's noticably cheaper but I don't think the picture holds up to them (and its 10.2gbs HDMI limited).
Sony Pros: Native 4K, good game mode, decent HDR
Sony Cons: Color banding (details below)
JVC Pros: Industry leading black levels, great game mode, decent HDR
JVC Cons: Not real 4K, e-shift process creates some image noise (mainly PC sources and text). Banding if CMD (frame interpolation) is on in a 4K source.
For movies, the Sony is incredible; I can't think of any compliants. For games, with 4K/HDR on there can be some severe color banding which can be super annoying. The running theory is that while it can accept a 13.5gbs signal (4K/60hz/10bit 4:2:0), it's processor is limited to 10.2gbs (4K/60hz/8bit 4:2:0). Basically, that means 10bit color gets downscaled to 8bit. Many times it's not a big deal, other times it looks just awful. This must be considered before buying a Sony projector. There are lots of photos if you search online.
The JVC does not have this issue, because it has a full 18gbs board. The JVC has its own issues, like e-shift artifacting and CMD (frame interpolation) causing banding with 4K sources. There are higher end Sony (885) and JVC projectors with laser sources that fix these issues, at a cost of over $25k.
So for 4K/HDR projector gaming at $5k to $10k, you either have to accept the issues and get whatever is the lesser of two evils to you, shell out $25k+, or wait for prices to come down.
For me, I picked the Sony because it is nearly perfect for movies, and I can turn off HDR if the banding gets bad in games. I couldn't get over the e-shift look/noise of the JVC models, since they are not really 4K. You will find many competing opinions on this in AV forums. Sony used to be well behind JVC, but the 285 at $5k has gone a long way to being price competitive with JVC. Note that the 285 is an upgrade over the 365, for half the price that the 365 launched at.
As a side note, all projectors handle motion differently. I prefer the "look" of a Sony projector, but this is just my personal taste. Also, AVSForum is a great resource for discussion of these projectors.
Hope this helps!