I will assume the you are applying your custom transition like so
(set: $transition to (transition: "pulse") + (transition-time: 2s))
This text should just fade in.
$transition[While this text should pulse for two seconds.]
1. Each time a Passage Transition (*) occurs an equivalent of a (transition: "dissolve")[...Contents of Current Passage...] is added to the queue of things to do once the HTML elements representing the contents of the current passage has finished been generated.
2. Your custom "pulse" transition is also being added to this same queue.
Once everything is ready the actions/animations contain within the queue are processed/activated one by one.
Assuming that the above two animations are the only things within the queue then the 1st outcome you see is the contents of the current passage "dissolve" (fade) in, and the 2nd outcome you see is the "pulse" effect being applied to your marked text.
Unfortunately the 1st ("dissolve") animation in the queue is also activating the 2nd ("pulse") animation, which is why you see that animation twice. I believe this may be due to both animations internally using the transition-in CSS class of the special tw-transition-container element. You can use the Animation panel equivalent of your web-browser's built in Web Developer Tools to see the sequence of animations during the rendering of the current passage.
Due to how Harlowe's Passage Transition Effect is implemented overcoming this issue may difficult, I personally can't think of a way to so so but maybe someone else will. I strongly suggest you create a new issue on the Harlowe project repositiory and let it's Developer know about this issue, they may know a work around or be able to supply a fix.
(*) The process of moving from one Passage to another, this other can be the same as currently being shown as well as the first passage of the story.