This idea was inspired by New Rep indicator stops when having negative score. Quick summary of that post: OP placed a bounty, decreasing rep by 500. After that, new rep increases were not shown by the top bar achievements indicator icon (until the OP clicked on it to view the details, which reset the calculation).
Apparently, the indicator is currently set up to show the difference between your current rep and what your rep was at the last time you clicked the indicator to view the details, but only if that number is positive (personally, I think we should Display lost/negative reputation in topbar achievements, but that's another story). So -500 rep from a bounty obscures all rep increases until 500 more rep is gained (or until the calculation is reset).
So my proposal is simply to adjust the calculation to ignore the effect of placing bounties. The rep change from placing bounties is really just an artifact of bookkeeping, and thus it doesn't make sense to even take into account for reporting positive rep changes in the achievements indicator.
Implementation of course depends on how the system is set up, but it seems like it would be a relatively straightforward task, however it works. There has to be something tracking the state of the indicator for each user. So when a user places a 500-point bounty, just adjust whatever is tracking their indicator state by the same amount. That way they'll continue to receive notifications about upvotes et al.
Don't deprive bounty-placers of the little dopamine rushes that seeing a green +X in the top bar provides. Hurrah for positive reinforcement.
As Richard Tingle points out in the comments, when the bounty is eventually awarded, that rep increase on the person who received it should definitely still be reported and included in the indicator calculation. It's just the loss of rep for placing a bounty that should not be included.