But, he went on "if you wear a respirator and had the windows open, the likelihood of a problem is near zero".

@Dwayne Poster I was going to wear a full respirator.

Clean the areas you can (while always wearing a respirator), trap the current infestation sing snap traps, then seal entry points and continue bait boxes on exterior.Since your property boarders a creek and heavy foliage, you will ALWAYS have a rodent issue so it is important to keep bait boxes along the fence line boardering the creek and keeping the population down and away from the building.

Methyl chloride strippers work faster but you need a full-on gas mask or SCBA gear to safely work around them -- a half-face respirator fitted with organic vapor cartridges, the painter's standard, won't fully do the job for long exposures, and you'd need long exposures to get all that trim stripped.

They seem to think that your only value is identifying properties on MLS.Before I moved to Maine, I had an investor (now a personal friend) who I ran around for hundreds of accumulated miles, showing many disgusting, abandoned foreclosures - while wearing a respirator because of all the mold - until some inexperienced knucklehead agent brought him a wreck that he could rehab. 

We use a respirator with two HEPA filters, seal the doorways inside the house, remove everything from the room, cover walls and floors with plastic and we wet the ceilings with a garden hose or HUDSON SPRAYER 4 to 5 times until the popcorn turns to mush.

This very same question arose yesterday.In short, asbestos/portland siding is not terribly friable (unless you are drilling holes through it) and can safely be covered over or removed (by yourself).If will be cutting holes in the wall, just make certain your workers are aware of the siding and they take appropriate precautions (a properly rated respirator. 

@Drew ClaytonI grab a tyvek suit and one of those dual sided respirators.

Should I just throw on a certified respirator and seal off the room?

I've got to say, the Kiltz original primer, even with a VOC respirator on and windows open, is pretty noxious.