By the way, Treasures Against Time is very inconsistent with how they PRINT the Master's name. They show it both ways, so the example of the letter you mention (page 184) is inconclusive.
Then it should be easy for you to show us some examples of where the title is spelt without the "a" present. Show us about half a dozen examples, (except in direct quotations where the original spelling is maintained for accuracy) page numbers should be all that is required for us to check for ourselves.
In fact the book always spells the title as Paramahansa except in direct quotations, e.g. the letters written by Yogananda where he has signed the name as Paramhansa. The fact that the spelling as Paramhansa is kept in such letters is one of the reasons why this particular letter is good evidence that the title was signed with the extra "a" on this occasion.
As for your claim "that's how the Master signed his name. Always." I'm sure we would all like to know how you can be so sure of this. Have you seen every signature he ever made, or are you just repeating what a certain former Swami has said?