Welcome!
If that is your DC's real name, you may wish to edit, as this is a public forum. If it is a pseudonym, never mind!
It sounds like your daycare provider is suggesting that she may have both a developmental delay and giftedness, which is why she suggested nonspecific testing. The most straightforward approach to assessing both areas at once would be to contact your local public school district, and request a special education evaluation, which will include testing for both cognitive and academic strengths (giftedness) and language and social deficits (possible disability, such as a communication delay or autistic-like developmental delay). A school district evaluation would probably not begin until September, since schools are not in session now. It would be at no additional cost to you (beyond your taxes), and would most likely connect you to school-based interventions and therapies, in the event that an actual delay/disability is identified.
A private/pediatrician-mediated evaluation would most likely include the same major areas of assessment, but may or may not be covered by your health insurance (if not, cost begins at roughly $2000 and goes up from there). Outside of cost, there are pros and cons to both assessment avenues, mostly having to do with the principal aims (special education eligibility determination/child find vs the family's expressed referral question), and also differences in professional orientation between schools and clinics.
In either case, you should be aware that cognitive testing at this young age generally is considered fairly unstable (both higher and lower than subsequent testing), and is compounded by the kind of communication and developmental delays suspected in this case, which often render children less consistently and accurately testable than otherwise. IOW, whether she tests as GT or not, that score may or may not hold up in future school-age or adult testing.
As you clearly indicate, the more important reason for pursuing evaluation is to investigate the possible disabilities/delays, and intervene, if necessary. Any info on giftedness should probably be considered a bonus, at this point.
