I'm assuming that your daughter is very young--not quite 1 yet maybe? Older children are usually capable of giving a urine sample, so if they had to use a catheter then obviously she's not old enough to do that, yet.
In small children, fever, vomiting, and general crankiness can be the only signs of a UTI. They can also be signs of gastroenteritis, which, at this time of year, is ridiculously more common in young children than UTIs are (my son had a 24-hour one last night/today, for example). If your daughter had had gastroenteritis, the doctor's initial recommendation would have been perfectly acceptable.
Unless you think your daughter is going to be having UTIs frequently, asking for her to be catheterized whenever you need to go to urgent care might be a little extreme--especially if it's obvious she has, say, an ear infection or strep throat. My children have never had to be catheterized for anything, but I have friends whose children have had to be catheterized for UTI testing, and, from what I understand, it's pretty miserable and somewhat invasive. In other words, it's probably not something you'd want to do unless you had to. I've had to be catheterized a few times and didn't especially enjoy it and I was an adult. But now that your daughter has had 1 confirmed UTI, it would certainly be easier to convince a doctor to test her for one if you find their initial conclusion unsatisfying and she has the same symptoms.
If your daughter keeps presenting with UTIs, she may need to be assessed by a pediatric nephrologist as there could be an underlying medical problem causing the infections. I've done some searching and I can't seem to find any reported connections between playing in the sand and urinary tract infections, but I'm not a doctor.
As for other tests/diagnostics you could request anytime you go to the doctor (not necessarily urgent care):
- X-rays: Obviously most commonly used to assess broken bones, but chest x-rays can be helpful if you/doctor suspect something like pneumonia.
- Blood cultures: For bacterial infections. They usually only like to do blood cultures if a child is fairly ill--it does involve drawing blood which can be difficult in small children. It also takes a couple of days to get results.
- Urine cultures: Similar to blood cultures, but for urine instead. If there is concern that the bacteria causing a UTI could be antibiotic-resistant or simply not respond well to standard treatment, doctors will sometimes do a culture to determine the type of bacteria causing the infection.
- Flu test: If you or your doctor suspect your child has or is coming down with the flu. This is especially helpful if you catch the illness early and your child can take Tamiflu, but you have to catch it within the first 48 hours of flu symptoms, I believe.
- Throat swab: Usually to test for strep, but a throat swab can also test for Mono. It's not practical or reasonable to perform either of these tests if your child does not have a sore/red throat or swollen lymph nodes. They're two separate tests, but I've had both of them run on me in a single office visit and had results returned very quickly (10 minutes or less).
Those are the tests I can think of that would be considered more or less "standard" offerings at an urgent care clinic or pediatrician's office. Other, more non-standard diagnostics, like ultrasounds, would probably need to be performed elsewhere.
As an aside, we've recently seen an up-tick in pediatric after-hours clinics in our area which are super nice because they only see children. Don't underestimate how awesome it is for you to go somewhere with your child that only specializes in children.