Should the home inspector do the pest inspection?

No, no, no.

Why?  Do you want a car salesperson doing a brake job on your car?  Would you take a cruise on a fishing vessel?  Do your DVDs fit in an 8-track player? (Okay, I'm really dating myself here!)  But, seriously, the concept of "specialization" exists for a reason and as a buyer, you need to rely on the best information possible.