How do I resolve this issue as it is causing a great deal of unpleasant feelings in the home?
There isn't much you can do to avoid ruffling a lot of feathers about this except to keep yourself calm and logical in your discussions with your wife and your daughter, and some of this depends on how you've all interacted before in your relationships. However, this seems a good time to either learn critical thinking skills or to put them into action.
There is a fine line between an 'appropriate' bikini and an 'inappropriate' one. First, you need to find out exactly what it is that makes this one so objectionable. Is it that your daughter bought it without her mother's knowledge? If so, that is what you should focus on, not the bikini itself. Had she tried it with her mother who said no, that she could not buy it? If so, that is what you should focus on, not the bikini itself. Is it that your daughter spent money making a decision apart from her mother? If so, that is what you should focus on, not the bikini itself. (I think you get the picture. The real issue isn't likely to be the bikini.)
You state in a comment,
...she does not have an issue with our daughter having a bikini and during the shopping trip there had been no issue of this garment being denied they had looked at a number and had agreed on the two that were purchased.
If none of the above reasons apply, then it might be the bikini. If it is, you need to discuss in private with your wife what makes that particular bikini objectionable. If it is truly a modesty issue, decide where it is logical to draw the line and why, then sit down together with your daughter and present your case. Then listen to your daughter's reply. She may want to have a more fashionable bikini. She may want to have a sexier one. She may want the autonomy to buy one herself. You can't discuss this reasonably until you understand what's going on in your daughter's mind. Then discuss the real issue.
If it's just that ties can come loose and cause an embarrassing situation, who is it going to embarrass? If the answer is only your daughter, instructing her to keep the ties well tied should be enough. If your wife doesn't trust guys to keep from making accidents happen, this needs to be discussed as well.
For example, if she wants to be 'sexier' and you think that's inappropriate, you might want to explore why a young woman her age should not be trying to "look sexy".
The bottom line (in case it's not clear) is that this sounds like an emotional issue right now, but it needs to be explored and the deeper issues dealt with reasonably. That's where critical thinking comes in.
Taking scissors to the bikini is not, in my opinion, a reasonable response, but a very dismissive and punitive one. If she paid for it with her own money, it should not be destroyed. If your wife objects to it that strongly, she should either (after a mutually respectful discussion) forbid your daughter's wearing it yet, or perhaps buy a replacement for it with her. That's much more respectful and less likely to get your daughter's hackles up.