Adding an instrument rating doesn't change the fact that your FAA private certificate is foreign-based. The IR is an add-on to a pilot certificate, it isn't a certificate itself. If you don't keep your foreign license current then you can't fly on your FAA foreign-based certificate.
To get rid of your foreign-based certificate, you can either do a regular FAA private checkride (including the written exam) or get an FAA commercial certificate (which replaces your private one). In both cases you'll also need an FAA medical to replace your foreign medical.
Source: I did an FAA IR while holding a foreign-based private certificate. I later did the private checkride to get a 'full' FAA private certificate. See this answer for a few additional details on going from foreign-based to 'full'.