A fireplace surround is a relatively small area. I say change it to what you want. Recommendation based on the paint swatch:

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Painted tile never looks good IMO. It's a very short term solution. Retile the whole thing if you want a change.

Actually, you can stain travertine. You must prep it properly and seal it afterward, but despite what you may hear it can look great. I have seen it done very successfully-but then, I'm not trying to sell you anything. It really depends on your own level of skill and how comfortable you are using solvents.

If you have done woodwork with pleasing results, you will probably have the skills to stain your travertine. Here's some good information to get you started (but there's more info out there which I suggest before starting your project) https://homeguides.sfgate.com/stain-travertine-34742.html