I agree with others that you should not push too hard on a finding employment elsewhere until you get your MBA. I don't know how much value that adds to you, but I have to believe it will do something and you should get more money if you switch jobs after getting your MBA, then if you switch now and then expect or ask for a raise after getting the MBA.
Where I partially disagree with the other responses is on not bringing an offer to your company as leverage. In some fields (YMMV) doing this is absolutely a valid strategy. It comes with risk but it can be very effective. The corporate world no longer expects employees to stick around forever and with all the movement that happened during the pandemic, countering is happening and is part of the game now. All the red tape of getting a promotion or raise can evaporate so quickly with an outside offer. I've seen quite a few colleagues ask for something, be told it's not in the budget, go get an outside offer, and within 24 hours they are offered a matching salary or offered a promotion. I even heard of a manager telling his report to go get an outside offer because the higher ups didn't have enough promotions to give out. The main key is you have to do it tactfully. It can't be "I went and got this offer elsewhere, match it or I leave". It has to be more of "A recruiter reached out to me with an opportunity, I cautiously pursued it and they gave me this offer. I really like working here and I don't want to leave, but the offer is really tempting. Can we discuss my compensation?"
But here are the considerations you have to make before doing this:
1. You have to be ready to walk. It they don't like you as much as you think they do or they just really don't have a way to address this and don't counter, then you have to walk. You can not stay if they don't counter because then they just caught your bluff.
2. Some companies/industries may look down on you for this, but it's way more common now and somewhat expected. It's not the red flag that it used to be. Yes, the majority of people that do this end up leaving anyways in the near future, but you have to do what is in your best interest. You are offering them to ability to continue getting value out of you. Sure if may not be for the rest of your career, but if they like you then hopefully they'd want you to stick around for another year or two even if you can't commit to 10.
3. You shouldn't bring forward any offer/accept any outside offer for less than 15%. I consider 15% to be the minimum. 20% and up is ideal. Depending on the job and industry, switching jobs takes some time to adjust and your growth may be slow early on at the new company. So the compensation has to be there to offset that. Or there has to be another benefit there which you company doesn't offer.
4. If you can ask for something other or in addition to money that really helps the situation. It shows that you aren't just out there to get any extra dollar you can. I know you said optimization, but optimizing isn't always about just money. For example, if you want more vacation days or days working from home, etc, asking for those too will help if they want to negotiate the counter.
dragonwalker, the piece here which still complicates all of this is that your current company is seemingly entertaining (maybe even actively working on) trying to get you into their management. If that is truly the case, then this may be one of those situations where they actually do want you around for the long haul and showing some loyalty (not seeking outside offers) to them could be beneficial.