Can this requirement be a sole-source to the NPO that has accomplished all previous phases? If not can it be solicited only between NPO's? Since an NPO is not a Small Business and requirements over $3K are automatically SB set-asides, what authority would be used to break the set-aside?
FAR 2.1 clearly specifies that "contracts do not include grants and cooperative agreements covered by 31 USC 6301"; therefore if you have grant money, you are not awarding a "contract" but an agreement. The FAR is not your regulation guidance, rather the DoD Grant and Agreement Regulations, DODGAR, should be the source for addressing your questions and providing guidance.
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Specific to your question, you should use the principles provided in DODGAR §22.305 General policy and requirement for competition. I also recommend taking the DAU online course Grants 201 or you can access the course material through the "student course material" section of the icatalogue.