What would happen in this situation?
Hypothetically, TNSE and ASG come to an agreed price of let's say 80 million. A local owner would've payed that but wasn't able to get ASG to come down to that number from $110 million, for whatever reason.
Would the local owner get a chance to match it before a deal was finalized?
It would be up to the League. If ASG & TNSE come to an agreement, it would be up to the NHL to approve the relocation or not.
There are no hard and fast rules under By-Law 36 (ie no requirement that a team MUST sell to a local owner) - just a list of 24 considerations that teams are supposed to consider when voting to approve a relocation, with no requirements how those considerations are weighted.
Two of those considerations are:
"(d) Whether the present owner of the Club has made a good faith effort to find prospective purchasers who are prepared to continue operating the Club in its present location and/or has engaged in good faith negotiations with such prospective purchasers.
(e) Whether there is any prospective purchaser of the Club and franchise who is prepared to continue operating the Club in its present location and, if so, whether any such prospecteive purchaser is willing and able, if necessary, to sustain losses during at least the initial years of its operations there."
If there exists a credible local offer, but ASG is demanding a higher price than a relocation offer, the League/Teams would have grounds to reject the transfer on violating the "engaged in good faith negotiations" consideration - but it/they would not be required to do so.
In that case, if the League wanted the team to stay, they could reject the relocation offer and make ASG negotiate with the local group, or if the League wanted them to move (or just didn't care, or just wanted this new soap opera to end), they would approve the relocation bid and they would.