What advice would you give for a way to make this entryway last longer without major concern of the potential moisture issues and smells that come along with constantly wet carpet?

Vines in the house compromise the frame by separating the siding which lets moisture in. 

The basement is made of stone wall which my inspector says is prone to moisture getting in which turns into mold.

Over the decades, "leaky air systems", bad plumbing, years of moisture causing rotting wood are "termite magnets" for older mobile homes.

One thing a good inspector will have is moisture detector, and temperature sensing equipment, that could tell you if something is leaking somewhere. 

Each day that the carpet was wet, it was treated with an antimicrobial solution and everything was tested for moisture and mold several times throughout the process.

The main things you want to look for or avoid in the area, in my opinion, is going to be Flood Zones, moisture issues in the crawl space (it's very humid), and maybe knob and tube wiring if it was built before the 1940s. 

The description says that the basement has areas which need mortar repairs, a leak in the chimney, and signs of moisture in the first floor bedroom, but you've allocated $0 for an initial repair investment. 

I had on a similar issue noted on one of my properties as a seller as there has been some moisture in a crawl in space and we gave a credit.

@Andrew SchraderRing Alarm Flood and Freeze Sensor https://www.amazon.com/dp/B07M943G7L/ref=cm_sw_r_sms_apa_i_ex3lEbD8K376BNotifies you of moisture or temperature drops.