I cycled for 3 months around Australia a number of years ago, and I went up to 5 days without showering or changing my cycling clothes ... depending on the availability of showers, laundry facilities, etc. I'm sure there were moments when I did not look or smell fresh and clean. But at no point did anyone treat me badly in any of the shops or cafes I went into, and at no point did anyone even hint that they thought I might be homeless.
I did, however, have a lot of people asking where I was from, where I was cycling to or from that day and over the course of the tour, and all sorts of interested and curious questions regarding my tour. So much so that I felt like I should have done up a small brochure which I could hand out!!

But all the people who talked to me, and there were lots, seemed to know that I was a cycletourist visiting Australia ... and that I wasn't homeless.
I've encountered the same situations in Europe and Canada ... lots of people who know I'm on a cycling tour and who are interested in the details. And
no one who has
ever made any indication whatsoever that they think I might be homeless.