First, about grout lines: not good to make them really thin. Second, tell us how thin a thin line is. If you line up 9 tiles in a row spaced as tight as you like the line to be, how much is the total distance? 55 inches? 54.75"? 55.25"? 55.50"? This will help the discussion stay on track when people say "thin" and then thinner or thicker from that measurement.
Your problem: The shape of the frame is designed to be curved in front. (Is this right?) The slight curve, a design feature, is significant enough to be noticeable. Big question: how much is this curve? The bulge in front. Which tub is this? Its shape will be visible in a pdf file somewhere on the manufacturers web site. Tell us more.
A curve is a bit more distance to cover than a flat surface. This extra distance has to compensated for by increasing thin grout lines into thick and thicker grout lines. Not good. You have very few options.
One extremely unlikely option is that you will find enough variance in tile size that you will be able to use only "bigger" ones on the front of the tub surround. Tiles are not perfectly square and not precisely the same length on all sides. If you hunt (select and deselect), you will find some that are slightly different in size, perhaps with a non-square shape, perhaps almost like rectangles. You could screen out tiles that are longer on one side from those smaller on one side and that might be a start to your solution. Seriously. Sounds like fun to keep you amused and planning for days to come, but it is serious, and some tile layers can check squareness and variance within a few minutes and then calculate options within another few minutes. However the variance on a small tile is a small small number, so don't get your hopes up, but do do the math nonetheless.
The other options all make use of some tiles in the layout being cut down in size slightly. Perhaps this won't look good because some tiles would lose the rounded edge and others not. This can be worked around easily. Many good looking tile layouts can be suggested to you, some of which require the rounded edge to be cut off on only a few tiles.
Perhaps your tile layer has no saw.