There are no legal implications. I am free to offer you a job right now, and if you don't take it, tough.
I am a headhunter, and the last time a client suggested that was how we ensnared someone we wanted, I spent considerable effort persuading them not to do so.
This wasn't ethics. Because my firm only deals with smart people my expectation was that it would blow up the deal, and I wouldn't get paid.
As a pimp, I have the objective facts that you're good enough to get a job offer, and an interview for a better job.
That means your "price" in the market is higher than what you are being offered. They know this as well, and that's why they are trying to stop you.
My targets are normally banks, so when counselling others who have been in a similar position is a counter offer.
I would thus get them to say "if you offer an extra 10%, I'll say yes now".
Part of the strength of your position is that interviewing staff can be a draining process for the employers, and for all you know they may not believe that the 2nd best candidate is anywhere near good enough.
Can't say if this will work for an LA, but maybe you could ask for the next level up on whatever pay scale they use.
It can put you in a better position in two ways.
Firstly they may give you more money.
It will also buy time, since they need to go away and think about it.
Thus I'd ring them after 11 tomorrow, and put your counterbid.