Yes, I would retake, and the case for it is fairly clean.
Start with the two programs. Your 645 sits about 20 points above the median for the other program on your list and about 30 points below the median for the Master's in Finance, which is the one you actually want. So right now your score is doing its best work for the program you'd rather not attend. Medians aren't cutoffs, and a 645 won't get you screened out of the Finance pool on its own, but it isn't helping your case there either, and 30 points is a very reachable gap.
The stronger argument is the gap between your practice and your test day. You were scoring 685 to 735 and came in at 645, which is 40 points below the bottom of that range and 90 below the top. Before reading too much into it, though, one question matters: were those official practice tests from mba.com, or third-party mocks? If they were official, the gap says your test-day execution came in under your actual prep level, and a retake has real upside without a rebuild. If they were third-party, calibration varies across platforms, and the 645 may be closer to your true level, which means the retake needs more content work behind it.
On the sections, Q81 is your lowest and the most obvious place to gain. Verbal needs to come up too. For a 685 or better, you generally want Quant in the mid-80s and Verbal around 83, so both need real study time rather than touch-ups. Your DI83 is the one section already at the level that target calls for, so keep it in steady rotation with small regular sets rather than rebuilding it. It's also worth noting that the Verbal strength you saw in practice didn't fully carry over, since your Verbal landed almost level with your Quant on the day.
For the work itself, go topic by topic rather than running mixed sets. Relearn the concepts in a topic, practice only that topic untimed until your accuracy is consistently high, and for every miss, identify the actual cause: a concept you didn't know, a misread, a careless error, or a trap answer. Redo those missed questions from scratch a few days later. Once accuracy holds, add timing, then fold the topic back into mixed sets. That sequence is what closes a 40-point gap and tightens the range at the same time.
So, concretely. If those mocks were official, plan on several weeks of focused Quant and Verbal work and book the retake once your practice performance has tightened enough that the low end of your range sits comfortably above 675 rather than resting on it. If they were third-party, treat the 645 as your real starting point, sit a free official test from mba.com to get a clean read, and build the timeline from that number instead.
This article covers the retake decision in more depth:
Should I Retake the GMAT?CoolCord
I took the GMAT today and scored 645 (Q81, V82, DI83). While taking mock tests, I scored between 685 and 735, being particularly strong at the DI and V sections. I am planning to apply to the Master's in Finance at SSE or IB at SSE, which have median GMAT scores of 675 and 625, respectively. I would rather study the Masters in Finance, should I retake the exam?