Yes it does. That's pretty easy to test and find out.
I checked my 'BMR' on a Report, weighed in with a significantly different figure without doing a 'save your goals' thing, and the Report showed a new BMR. It'll be calculated at any moment, from the most recent weigh-in.
However, Heather brings up the reason for the discrepancy. Spark will always show about 300 more than other places due to the multiplier. Note the report (daily calorie differential) says:
The BMR in the chart above has been multipled by 1.2 to account for the calories you need for both basal metabolic functions and light activity outside of exercise (such as light walking on the job, deskwork, and running errands.)
Deb, in New Zealand