It would not bother me, but I do get how it might bother some. If both metals are polished, you really won’t notice the difference so much when wearing as the light reflection will kind of obscure the differences. If you have a satin/brushed finish, the colour differences will be more apparent as those finishes tend to bring out the colour more. This also applies as the ring scratches over time and take on a more satin-y finish, but the head and top of the ring tend to be more protected from scratching (compared to the underside/palm side).
I am not sure if the head on that ring is platinum or white gold?
As some said, part of it is a colour issue so some jewelers prefer to use white metals for heads/prongs – the white blends in better with the stone and prevents the gold colour from reflecting off the stone; not usually as much a problem with ideal/excellent cut stones where most of the light comes from top and reflects back properly, but can be more of an issue with poorer cuts.
But aside from that, it is fairly typical to have gold (of whatever colour) rings and, say, platinum (white) prongs for better protection of the stone. This is because some gold alloys are more susceptible to being TOO hard and/or brittle, whereas platinum tends to be softer and “bend”, so if you do bang your ring or something, you are less likely to lose the stone with platinum prongs/head which will just bend and keep holding the stone. A harder and/or more brittle alloy in the prongs/head though may instead break, and the stone can go flying, or bend back and leave the stone loose. If you bang your ring and have prongs you should go get it checked out anyway, even if it looks fine, but the problem is sometimes people unknowingly bang it or do think it looks fine, when really a prong may be damaged, the stone loose, and they may lose it.
I have also heard some jewelers say rose gold can be particularly brittle (because of the alloys it needs to appear “rose” – copper does not like to alloy with gold and has to be kind of forced into it, and part of it is also related to production process i.e. cast v. hand forged, hand forged metals tend to be harder than their cast cousins and have less impurities/risk of brittleness). So I can see why it could be prudent to use a different prong for a rose gold ring.
That being said, if it is going to bother you, and it sounds like it might, you can contact Brilliant Earth and see if they could make the entire ring for you in rose gold (there may or may not be an added cost), or you can look around for a stock setting from another vendor that is all rose gold.
Whatever you get, do get the prongs checked regularly by a local jeweler. Prongs do wear, and sometimes you may damage them without even knowing it, increasing risk of losing your stone.