The Tyvek in your case will be your "Water Resistive Barrier" (WRB). It is critical that the WRB be installed with great care if you hope to avoid water intrusion in the future (as well as effectively air seal your home) as this is your real line of defense against it. This means paying attention to your overlapping seams, taping all seams and sealing it with diligence at both the top and bottom as protection against wind driven rain.
IMO, a better choice than Tyvek are the current generation of self adhesive house wrap membranes. The one I prefer to use is a product by Benjamin Obdyke called "Hydrogap", which also creates a drying space behind your finished exterior (shingles, claps, LP Smartside, Hardieplank, etc). It is not a specialty product and is widely available (in stock) here in the Northeast. There are similar products available from a number of other manufacturers. The big advantage of these products (IMO) is that they eliminate the need to drive fasteners (putting many, many extra holes) through your WRB and it makes installation easier and less $$... and no plastic "buttons" required (!)
I have used a number of synthetic siding materials on homes in challenging climates and locations (coastal direct waterfront most notably) including LP B&B, LP clapboards, Hardieplank, Azek and all the common natural wood products in various species.
From my experience, Hardieplank, applied correctly, holds paint amazingly well - we've never had any peel. It is a bit unpleasant to work with (cut) as it creates a lot of dust which should not be inhaled so protection is prudent. It is also heavy and some care must be taken in carrying longer lengths so as not to snap it.
We have used LP B&B, claps and t1-11"esque" forms. It holds paint well, is available prepainted for a very low upcharge (highly recommend this if their colors work for you) and is as easy to work with as any similar-form wood product. It is also available in 16' lengths, which is nice if you have longer runs and are trying to minimize seams. B&B (in general) pay particular attention to seam caulking to avoid trapping water which can cause big headaches down the line. We have used Level clear caulk exclusively for many years - it is amazing stuff and we'd never go back to silicone caulks - We've used this for 5 years now and have experienced no issues with its durability. We are finishing off a lakefront house right now using repainted LP B&B.
I have a real dislike of Azek principally due to its expansion/contraction characteristics (and price!). You need to leave big gaps at ends and it will always look sloppy as a result. And in our climate, invariably we get some occasional cracking/splitting between nails which obviously looks horrible and requires expensive rework to remediate. We prefer to use a product called Boral, which doesn't have these issues and is probably more durable long-term. It is bade of fly-ash in a binder and does not rot. Our crews vastly prefer working with this to working with Azek and similar. Care must be taken when carrying longer lengths as it can snap.
Hopefully, some of this information is useful to you - good luck with your project!
All the foregoing is just my opinion based on my experience, YMMV
