Vinyl siding is generally not recommended for exterior chimney siding because of the heat that the chimney produces.
How to install vinyl siding on a chimney chase.
How to install a chimney chase cover by rockford chimney supply.
The best solution is to contact a certified chimney sweep to inspect your chimney and determine the root cause of the problem.
I have looked around and a stainless steel chase cover is around 350 500.
If the wall on the house side of the chimney chase is insulated then you need none in the chase itself.
Set up a ladder to install furring on the higher portion of the chimney.
If there is not then you can install it but keep it from falling against the chimney liner as it could be a fire hazard.
Space the masonry screws about a foot apart and leave at least 1 foot between the two furring columns.
As far as adding insulation it kind of depends on the construction.
Especially if faux stone or brick are used but also if vinyl siding or other siding is not properly installed and sealed.
In this video marc leconte of high quality home maintenance shows the final repair before painting of a chimney that needed new wood siding to replace rot.
Chimneys are designed to withstand high temperatures and protect the structure of your home including the siding from damage.
Through the chimney chase final covering.
Siding on the chimney should never be hot to the touch when your fireplace is in use.
Builders installing a stone or brick veneer often cover the entire chimney chase exterior with a stick on flashing membrane such as wr grace ice and water shield.