Hello. My twins girls have just started at the local grammar. RedVW is absolutely right, "Having two children taking the test is quite intense" and then some!
Our approach was to be realistic with our girls and discuss with them the number of candidates vs the number of places - if they wanted to sit the 11+, they should be aware that there is no guarantee of getting a place despite being academically able. But given that we live within walking distance of the grammar and both my girls were keen to try, they gave it go. We are not the kind of family which would have them sit multiple tests at various schools just for the sake of getting them into a grammar school (I make no judgement on that, just saying it wasn't our cup of tea). If they didn't get a place then they would go to the local catchment comp. We talked a little bit both together and individually about the chances of getting into the grammar, the possibility of only one child getting in, whether that child would go if their sister didn't get a place etc. All the permutations of outcomes. I don't know how helpful any of that would have been, for us it was the separation (actually the not-doing-as-well-as-my-sister) which would have caused the most problems I think.
We paid for a tutor to help them, an hour a week individually from Feb until the test in Oct (with several weeks off when the tutor or us were on holiday). I was not confident to do the prep work myself and my relationship with both girls would have suffered severely if I had been the one telling them what to do.
The tutor did a lot of test familiarisation but also a lot of English and maths. She really helped shore up and extend their general knowledge and confidence.
Both my girls are similar to how you describe your DD, they were very motivated to prepare for the test. However there were a few wobbles during when they didn't want to do the homework: our response was ok, it's your choice whether you do the homework or not. But in our and the tutor's opinion if you want to stand a good chance of getting in, you need to do this homework as preparation. You choose. She chose to do the homework (eventually).
That might sound a bit cruel, but there was no way that we were going to stand over her shoulders watching her do a paper to ensure she did it. If her goal was to sit the 11+, then we supported her as much as we could by providing her with the help and papers etc. We had to remind her of her choice and let her choose again, along with the consequences of that choice.
OP are your concerns more apprehension that your DS won't do the prep work? Or apprehension that he won't get a place? Or actually the fear that he will compare his results against his sister's and possibly be upset with that? There's a lot of anxiety for us parents twins going through this!!
Because the pressure is immense for twins IMHO. Whether it's obvious or not (and only you as parent can really know), the comparisons that the kids themselves make between and against each other, it's constant and sometimes soul destroying. I think it's also unavoidable, because they don't live in a bubble as an only child. Probably similar to siblings, just intensified for twins. Especially between twins with similar abilities in some thing.
Presumably you have done / will do the open evenings. That might work for your DS to motivate him towards the grammar. Seeing the alternatives and being happy about those would relieve some of the pressure of "you must get into the grammar". If your DS says he would like to try for the grammar, be clear with him that it's his choice and this is what he needs to do to prepare himself and put himself in the running. If he wants to spend all his time playing football instead of prep work for example, then he will lessen his chances. But perhaps he will start with his tutor and that will encourage him through the prep work.
But we stated categorically to our girls that getting a place was not the end all and be all, that they were not changed or diminished if they didn't get in. The system is not perfect and a 2 hour test is purely a snapshot of a child in that situation, with a lot of luck thrown in. They wanted to go to this school, so they tried their utmost for that opportunity. I couldn't have pushed them towards it against their will. We gave them a small present after the exam as a reward for their effort, not for their results.
We are now finishing week three of secondary and the shine and excitement is beginning to wear off a little. But one of mine said already that it was really nice to be in a maths class where she didn't feel bored waiting for the rest of the class to finish, they all worked at similar rates together as a class. She liked the togetherness and not feeling "odd" or standing out for being good at maths. So that bodes well I think. We also have a few pairs and half pairs of twins in our year (and more in other years).
Good luck on your 11+ journey. (Apologies for such a long response.)