susan, ... Now you know the offer exists for SPG guests. If I needed a couple of "bridge" days and found a good offer on the SPG web site and verified it at the resort, I'd get in touch with SPG to make the reservation.
Assuming the rooms allotted to the offer are not all taken, you'll get your reservation and the kids can eat for free. And, if you have not ruined your relations with the front desk and it's possible for it to jiggle things around, you may even be able to stay in the same room.
The front desk at Lagunamar is not where you make SPG reservations. The clerks there are trained only know one job - how to check in and check out guests. Out of the US, it may be impossible for the people at the front desk to know all the details of all the types of stays they process. It might not be part of their trainig or responsibility.
An SPG reservation is not an SVN or SVO reservation - or an II reservation - or a direct LMR rental. The salad of acronyms denotes different corporations.
The days in the offer probably are ones from weeks retained by Starwood (SPG) or unreserved weeks released within 60 days of arrival for Starwood (SPG) to rent as it wishes. The usual limitation for resort stays is that one offer cannot be combined with any other offer. Starwood (SPG) has the right to make whatever offer it wants for the time slots Starwood (SPG) controls.
(As an aside, I would like to know if the resort shares in the restaurant revenue [I think Starwood runs that operation and keeps the profit, if any] or if any reimbursement to the resort for "free meals" for kids was negotiated away based on the economy.)
Since the SPG offer suits your needs and you like it, grab it! And, enjoy your 3 stays at Lagunamar. ... eom