Fabricating enclosures that look nice on your desk is its own art that will take time to perfect. Yes, you COULD husk some stuff like your Sonys for components, but IMO you're best off mixing and matching drivers from Parts Express. Rarely do low end consumer electronics perform well outside of their cost engineered intended use. Keep the Bluetooth speakers as they are, for their intended use.
DIY
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If you want to do this then why not. I’d go one of 2 ways.
- open it up and work out the size of the enclosure, half it and build 2 boxes with that size. Not sure how you would work out the port though so I would probably go with option 2
- remove the 2 tweeters and mount them in small boxes on your desk and leave the rest as it it (filling tweeter holes) under your desk.
Either way you will have speakers that play sound of some sort and If it doesn’t work as you hoped then at least you’ve had fun, and are a bit more experienced for your next go.
In my experience of building speakers from knowing nothing and just making speakers that look nice (no measurement at all) to actually using the speaker parameters to design proper enclosures and crossovers, they have all sounded ok. The later ones do sound better but my old ones didn’t sound bad.
If no one else offers any better advice, I say just give it a go and see how it turns out. 👍🏻
You best bet is to invest in buying / building custom stands to hold that speaker at your desired height over your desk.
The woofers will need the same volume of space either way to play according to spec. The tweeters probably don't get low enough to hide the woofers (more likely mid-full rangers) in a separate enclosure without some serious loss of mid-range from the woofers and bad directionality. While the home theater standard is satellites (tweeters / mids) playing down to 60-120 Hz with the THX standard requiring 80 Hz or lower, nearfield PC speakers can get away with 150-200 Hz as the low knee on the satellites without upper bass being too thin due to directionality from the separate sub.
It is unlikely you can fabricate an enclosure or set of enclosures that will sound better than the carefully designed one for those drivers. Just about every boombox and portable speaker is very cheap drivers with a very specific enclosure, DSP, and target listening volume. This is why so many cheap speakers and most boomboxes and outdoor speakers sound like ass at low listening volume levels. Heck, even a lot of the cheaper Dayton stuff even sounds like crap at low-mid volume or nearfield use.
As for the DSP, even with experience with DSPs and identified components, it's unlikely many would be able to reprogram it to re-balance for a different set of enclosures. With no experience with electronics I am going to outright say it is not going to happen, especially if you are asking basic questions without enough research and knowledge to be past those questions. The DSP is programmed against the drivers, enclosure, amp, and use case requirements and limits. It's not a chip that automagically fixes all that stuff.