Thanks All for your considered responses. I've read in detail and found a lot of helpful advice.
To be clear, DeWe, the lessons are supposed to be very play-based and tailored to a 3-year-old's concentration level. The half hour is broken up into chunks and includes clapping rhythms, dancing to the teacher playing the violin, naming the parts on the violin, teaching mummy how to play (violin isn't my instrument), and various other games. I'm present throughout.
In theory it should work - sometimes - and some days are better than others for sure. What Reallytired said resonates quite a bit: she's perfectly capable of concentrating for long periods of time on something of her choosing and in her own time, as opposed to a fixed slot. Her nursery teacher once described her sternly as, "very focused and intense in the way that some perfectionists who want to please everyone can be" and then looked at me inquisitively for signs of genetic traits
She's right about the ability to focus - and DD certainly thrives on one-to-one attention - which is why I thought Suzuki would be good for her!
Sometimes I wonder if it's my presence that causes the problem. But it's contrary to the Suzuki philosophy for the parent not to be present. Handmade, we're not doing group lessons as yet as I've been waiting to see how individual lessons go before committing her to anything else. I see the potential benefit to be gained from the social aspect, and with that in mind her teacher has already suggested she be given more chances to observe other children playing. I'm sure she'd like that.
Ohmygosh, i've heard about the box and ruler method of teaching! DD's teacher doesn't seem to subscribe to that method, thankfully; I think there's some flexibility within the broader Suzuki concept. My understanding is that it's based on the notion that very young children aren't to be trusted to respect and look after their instruments carefully so this was a way of minimising damage? Perhaps i've got that wrong? But the contrary theory is that allowing the child to hold and play the "real thing" should foster a pride of possession and encourage them to want to play more. I definitely see DD's pride in owning her very own instrument.
I generally only suggest practice, after dinner and before bath, when I sense she's got the energy for it and is in a cooperative mood. Practice is never more than ten minutes and definitely not every day. DH is very good at shouting out the occasional "remember to keep it fun" when he hears the tension mounting in my voice. I usually call it a day at that point.
As Pianomama points out, IMO it's crucial to avoid unpleasant associations at this age. I agree with many of you that life should be primarily about fun and play at 3/4, although I loved your idea of "meaning business" when she's 5, Pianomama 
Zen, you do talk a great deal of sense, to me, it's just that the Suzuki philosophy seems very strict on technique, stamina-building (especially for violin and bow hold) and instilling discipline. I was a cellist, and generally considered to be fairly gifted musically. I could make a nice sound and play with lots of expression, but I completely lacked discipline and my technique was decidedly ropey. My teacher was probably too soft with me. She admitted as much herself, but I wasn't going to be the one to suggest a change! I think she inspired me in many ways but I don't know if she quite managed to teach me to be my own teacher, which, as Pugs points out, is pretty important - eventually - as far as practice is concerned.
Flussymummy, I hear you as far as the tough love issue goes! The teacher used those very words with regard to the ploy of letting DD believe we were leaving the lesson (with her in hysterics) due to her uncooperative behaviour, and it sent a shiver up my spine. Just as you say, it's so at odds with everything else I do parenting-wise and absolutely not what I wanted for my 3-year-old child when we started out on this journey.
So, taking everyone's advice on board, my first step is to stop worrying about attainment (your words, Ohmygosh) and see if things improve over the next couple of weeks. A lot of you have suggested giving it a break, which I now plan to do over the course of the summer, and then perhaps we'll see if she's more ready for it as we move into Autumn. She'll be 4 in November and a couple of you seem to have had more success with 4-5 year olds, albeit that they're still just babies very unfocused. (Sleepingbunnies, my DD breaks off to hug me up to twenty times per lesson!). If things haven't picked up a bit by then I'll certainly look into the other options suggested here (Kodaly, Stringbabies, etc.) although we've been doing "fun" music classes, in the form of Rhythm Time and other more informal singing groups, since DD was very young.
Watching and waiting - but definitely no longer worrying. Thanks again for your input here everyone. Thought provoking and helpful.