I assume your reluctance to use the non-chlorine bleach (usually called color safe bleach on the boxes) is out of fear of damaging the garment. You must consider that a swimsuit is designed to go swimming, and swimming pools usually contain chlorine. This is the principle ingredient in regular bleach and is pretty aggressive.

Also, garments sold in the US must meet certain standards for use and care, which include having a label with care instructions and standing up to those care instructions. If the label says use non-chlorine bleach, then you should have no reservations about doing so.

My bigger concern is why the yellowing to being with. If the suit is older, then this can be expected, and may not be reversible. If the suit is new, however, this seems to me to be a manufacturing defect and you may want to consider returning it for a refund. A reputable drycleaner or the International Fabricate Institute might also be helpful to you in this case.

As to your specific situation, I can only think of two possible explanations:

A sun/heat reaction to suntan oils or other personal care items. A suit not manufactured to US standards.

I would try a machine washing of the suit to see what you get. It does not sound to me like you will get a lot of improvement, but it is worth a try.

Your caution about using bleach is warranted. Typically, chlorine bleach is to be avoided on nylon, spandex, etc., common fabrics for swimsuits. However, in your case, it may be the only thing left to try. What does the care label say?

I'm assuming that the suit was sold via a US retail outlet? If so, you can probably rule out the manufacturing quality issue, but not with complete certainty. It might be worth your time to stop in and ask the store where you bought the suit if they have had any other issues.

It is possible that you have a color bleed from the lining, even though it is a different color. Many dyes experience pigment changes over time, when they bleed, or even as a result of a reaction from some catalyst (heat, light, perfume, chemicals, salt, - could be a lot of things). But again, for this to be the case, then there would have to be a manufacturing fault.

You might also try taking the suit to a reputable drycleaner, or avail yourself of the "Ask an Expert" utility on the International Fabricate Institute. They may have more technical knowledge from the fabric standpoint than I do.

You bet I've got an idea. RIT (the dye people) have several products to keep things white. The type you would want would depend on the fabric content of your suit. So find that on the label (probably nylon, lyre and spandex) and find the right type for your suit. Follow directions and you'll keep it white! It's always good to wash and rinse your suit after each use.

Just while you're showering wash and rinse the suit. If you want to keep it in really good chap and keep the chlorine in check, make a solution of water, cheap laundry soap and aquarium de-chlorinator (this is gotten at pet stores-get a small bottle and add about 2 tsp. to a 8-12 oz bottle with the water and soap).