Reusing reference letters

I am a first-year PhD student applying for STEM fellowships this fall (NSF GRFP, NDSEG, Hertz), which I applied for last fall as a senior undergraduate. Although I was not awarded any of them, the reviews from the NSF GRFP included a brief statement that my "letters are good". Hence, I am planning to ask one of my previous writers (a supervisor from a research-oriented internship) if he would be willing to provide a reference again.

Since he has communicated in the past that time can be tight for him (as I also observed when working with him), I was planning to tell him that he could reuse his previous letters (or make minimal modifications), since I believe that the letters were strong (he also wrote one for graduate admissions which resulted very favorably), and then hopefully it would not be too much of a burden to submit them.

I have considered the idea that his message about time could be his way of implying that he would not like to write another letter, as one of my requests for a letter around that time received no response, but given that we’ve communicated very positively since then and that the working relationship during the internship was very positive, I am hoping that is not the case. I figure that if he is not interested in providing a reference this time he will let me know or simply not reply, but in the case that it really is just an issue of time, then knowing it would be less effort than before could be a deciding factor in his willingness. I also considered asking other writers instead, but I do believe that his reference would be one of the strongest if he is willing to submit it, so I figured I should at least ask.

In light of this, my question is: Is there any reason to believe that reusing a reference letter for doctoral STEM fellowships (NSF GRFP, NDSEG, Hertz) would weaken the strength of the application?