As orthocresol already commented, apart from some nitpicks, the general direction is correct.
The only issue really is the addition of bromine, which does not occur via a [2+2]-cyclo-addition (2nd from below in scheme).
Ordinarily you would expect addition the addition of bromine to a π-bond to occur first via a polarisation complex and then via a bromonium ion (bottom of scheme). In the case of the enolate, the π-bond is already readily polarised, and sufficiently asymmetric that a direct addition will occur. The following depicts the HOMO -1 calculated at the DF-BP86/def2-SVP level of theory, the HOMO is an (delocalised) in-plane lone pair of the carbonyl-oxygen atoms in the C2v conformation (1st row). In the Cs conformation the π-bond is the HOMO, it is also the low-energy conformer (2nd row). (On mobile the rows might differ.)
Please note that the following scheme is just a proposed mechanism, and as such probably a reasonable simplification; the actual one might depend on more factors like solvent, temperature, pressure, etc., and might have more, or even less steps, other intermediates (ion pairs), hydrogen abstraction might be via water/base bridges, etc. For educational purposes I think this is sufficient though.
See for example Aromaticity-Bromonium Ion, Mechanism for bromoetherification of olefinic alcohol, Why does cyclopropane react with bromine?, What is trans-addition? and more...
Thanks to orthocresol pointing it out in the comments.
Enolate reaction with bromine (as opposed to ordinary alkene) is traditionally taught to not go via a bromonium ion (img), presumably because the electronic character of the two doubly bonded carbons is very different. But I have not consulted any literature, this is just based off undergrad teaching + textbooks, so...
Since I was unlucky to find any specific literature on the topic (I did not search very thoroughly), I have tried optimising a bromonium analogue at the DF-BP86/def2-SVP level of theory. I have been unable to obtain a stationary point (no local minimum and no transition state). Which is not unexpected given the HOMO/ HOMO -1 is strongly polarised.