Its 'total value' is equivalent to other uncommon items
The value of magic
The 'value' of mithril plate mail (or admantine plate mail, also uncommon) is undoubtedly more than 1500gp. But looking at the value of just the normal item without the magic bonus is a bit of a red herring; it is not really the comparison I think you need to be making.
Take the broom of flying - also on the uncommon list. It has the value of a normal broom (minimal) plus the value of being able to fly - effectively having a permanent third level spell effect and a huge tactical advantage. What would the 'cost' be for a character to obtain unlimited scrolls of fly? Certainly more than 1500gp. So the comparison you should make is not, 'how much is the non-magic version of this worth', but 'what is the total tactical value of this compared to the other magic items on the list'?
Little value added for most builds
What is the value added of the mithril magic beyond a normal suit of plate mail? Not much - no disadvantage on stealth checks and no strength requirement to use. In a typically class-balanced party of 7th level, the advantage to the person wearing plate armor of no disadvantage on stealth rolls is minimal - they are not going to be the one sneaking most of the time. The advantage to the person wearing plate armor of not having a strength minimum is minimal - by 7th level they will more than likely already have a strength of at least 15. If the armor said "you can use this without having proficiency in heavy armor" - now that would be huge because it would open it for exploit by classes that normally don't have that good an AC. But the actual 'magical' properties of the armor will have little to no effect on your game. This is not unbalanced. It is not a loophole. It is a reasonable choice within the parameters you have set.
A fair comparison
Consider a player who chose non-magical splint armor, with a cost of 200gp, within your budget, and for their uncommon magic item, a shield +1.
Compare that to a player who chose mithril armor and a non-magical shield.
Both these characters will have an AC of 20. The only difference will be that the mithril player avoids the strength limit and stealth disadvantage, which are unlikely to come into play in any event. If you think the splint-and-magic-shield character is fine, there is no reason to think the mithril armor character is unbalanced.
What about a high Dex fighter?
In the comments, the OP explains that the player in question is playing a high-Dex ranged fighter. In this particular case, the value added of the mithril is considerably more - both the lack of penalty to Stealth and the lack of strength limit are relevant and specifically beneficial to that PC. It is clear that this particular player is astutely selecting their magic item to maximize benefit to their particular PC, and that they are aware of it, by asking in advance whether it is unbalanced.
In this case, I would say that it is a loophole, and is pushing the maximum benefit, but is still not unbalanced when fairly compared to other choices and uncommon magic items, principally because the mithril armor is still heavy armor, and thus removes one of the primary benefits to their Dex score, the bonus to AC. Consider that this PC could just as well choose non-magical studded leather and a cloak of protection, also an uncommon magic item.
If the PC's Dex mod was +5, this would result in the exact same AC and better saves, a strictly superior choice. Even with a Dex mod of +4, there is now a tradeoff between the 1 better AC with the mithril and the 1 better saves with the Cloak.