Understood.
Do you have known vision problems?
A glaucoma test is not normally part of the flight physical. I've had one done once, as part of the flight physical, when I was a teen; I never went back to that AME. He was an eye, ear, nose, throat specialist, and was making up his own exam...back in the day when a big cup got placed against the eye and pressed. I've never seen another AME administer one.
I am not a physician, not an AME. I cannot give medical counsel, though I've had a lot of medical exams (the rough equivalent to staying in a Holiday Inn Express the night prior). Eye exams in a typical FAA physical will be the same for the Class 1 as what you've had for the Class III. The standards are slightly different, but if you have no problem passing the Class III, you shouldn't have difficulty with the Class I.
I'm at a point where my eyes aren't what they were. I'm moving up the chart to bigger letters; I used to shoot through the bottom line, no problem. Not so much now. I still hold a 1st class medical, and don't forsee that changing. If you know your AME and trust that person, it's a good place to start. Most AME's are not out to hang anyone; they understand we're working, it's a basic check, nothing more. Most ATP's will differentiate between a "good" AME and one who is not, and will go with the "good" ones. Local recommendations count.