Plenty of people record studio vocals with stage mics. If it suits your voice, your technique, and your room acoustics I wouldn't worry about it.
That said, the Beta 87A isn't the quietest of mics — it generates about 15-18dB more self-noise compared to a decent LDC — so noise might become an issue in gentle, exposed tracks.
Technical Editor, Sound On Sound...
(But generally posting my own personal views and not necessarily those of SOS, the company or the magazine!)
In my world, things get less strange when I read the manual...
The track sounds fine to me (very nice version IMO). You may sound a bit better with another mic, but you've got to find that mic first. All mics sound a bit different, so it's impossible to say how an LDC would have sounded in comparison. Some will be warmer, some brighter. Remember that performance comes a long way ahead of the equipment.
Now if you want a reason to go and buy a U87, or a very good clone of it, then I'd just go and do it. If you want to try an LDC, and you cant borrow one, then maybe a Rode NT1, which is good and has very low noise but is not too expensive.
I'd say that sounds pretty good. Possibly a slight brittleness in the high end but it is difficult to say for sure with Soundcloud because it mangles the subtleties in the sound quite a bit. If you really want to find a better mic, I would suggest hiring a real U87 or possibly a Fet47 for a day and comparing the results. That will give you a good baseline to work from. However, don't expect the results to be drastically better as you already seem to be doing well with the Beta 87a.
I've got one that I use on stage whenever possible (I think it works better for my voice than the usual SM58s) but I've not used it in the studio since I got an LDC.
Two reasons mainly, the self-noise that Hugh mentioned, and secondly having an omni LDC means I can get a bit further away from the mic and not have to worry about trying to manage / work proximity effect. Which just makes the process a bit easier.
That's a lovely version of the song.
I do use a Beta 87a at home for male vocals (baritone, rather than tenor).
I find the mic has a very appealing, almost old sound. The top end is smooth and unhyped (unlike many high output dynamics).
Being a supercardioid does provide more isolation than an LDC worked at greater distance (Are you recording vox and guitar at the same time? The 87a will provide an advantage).
I don't find noise an issue because it's obviously a mic to be worked close (2 inches works for me) receiving a healthy level of lead vocal.
The one thing I notice - Shure say the Beta SM87a is focused on the upper mids for clarity (I agree, but it's in no way harsh). As soon as I started using it, I notice a slight lightness in the lower mids, which is where the body of a voice lies. So it can lack the shunt of a good dynamic mic and is perhaps best suited for higher melodic singing (I recall it being Thom Yorke's stage mic of choice for a good while). TBH - you have a good mic providing a good vocal sound. But you also have a really good voice. Wonder what it would sound like through a U87...