It will come down to how much flexibility Stamkos will have, but there are several ways that JBB can make Tampa's offer more competitive.
The team as it sits, has 11.715 available(provided a cap of 87 7 next year) with 9F, 6D, and 2G signed. If JBB wants to keep Stamkos, Duclair, and upgrade somewhat on defense, it's difficult, but not impossible. First, Sheary has to be moved asap, which increases available cap to 13.715 with now 6 spots needed. Secondly, Perbix needs to be moved for whatever positive value, if any, that JBB can get. This increases cap space to 14.84. Finally, JBB needs to take the L and move on from Jeannot. Yes, he brings physicality, but for an oft injured, no offense at all 3rd/4th line type, the space needs to be used more effectively imo. This brings the available cap space to 17.505 with now 8 spots to fill to reach a 22 player roster.
Elevating Crozier/re-signing Fleury(or another another min level #7 dman), Motte, Chaffee, and 2 additional 4th line/13th F types should leave the team with between 13 and 13.5 million left for Stamkos, Duclair, and a dman upgrade. This allows JBB to offer Stamkos in the 6.5-7 range perhaps, Duclair in the high 3's to maybe a ceiling of 4, and a dman in the ~3 range or a little more. There us some flexibility available with these numbers, and it gives the team a good shot at addressing 3 big spots in Stamkos, additional top F and a blueline upgrade over Perbix.
Of course, if they want to leave the defense basically the same aside from Serg returning, they can keep Perbix and still have 2-2.5 million available(after the other lower cost signings) to use towards the higher end re-signings or a boost in the bottom 6 or blueline. So ultimately, moving on from Sheary, Perbix, and Jeannot gives them the best shot at getting what they probably want, while only moving on from Sheary and TJ is also a viable plan. Only moving on from Sheary while keeping Perbix and Jeannot leaves the team just over 10(after filling in the lesser role spots) to sign Stamkos and Duclair. While not impossible, there is much less wiggle room in negotiations, and there is no room for any meaningful add to the blueline.