Well, I think the benefits of your ds being in a multicultural environment still stand.
My advice would be not to panic and don't think of it as an all or nothing approach.
Firstly, I'd try and identify specifically what it is that you are concerned about for your DS and think about how to address it.
Relationships outside school - could he do Beavers, drama group, sport class of some sort to build up relationships. Does he have a particular school friend he would like to invite for a playdate. Maybe you could make the first move in this and maybe school could help with communicating with the other child's family if English levels are a problem.
Educational enrichment? I can understand that a school that is so stretched with ESL students and a constantly changing roll may not have the capacity to stretch brighter students as much as some others. What can you do to enrich things? As a tutor for many years I'd suggest the following:
Eat as a family as often as possible. You'd be surprised at how much children's vocabulary and understanding of the world grows when they are habitually in mixed adult/child conversation. It is the regular day to day interaction that makes the difference.
Keep reading to him from a wide variety of books both fact and fiction. Again the amount of vocabulary, grammar etc that children absorb from this is amazing.
Both of these, done habitually will be making such a difference although if you don't see the contrast with children who don't do these things you might not realise.
Then of course there are the traditional enrichment activities. Take him to museums and historical sites, age appropriate plays, cultural events (Christmas carol service or harvest festival, Chinese new year parade, remembrance Sunday event, family friendly pride events, parkrun or other sport event, etc).
On top of this make sure he has opportunities for spatial and mathematical learning (weighing and measuring for baking, lego and hama beads, scooting and bike riding, water play in bath - he needs containers and jugs to pour and get an instinctive feel for volume and capacity- , play games involving dice and numbers. Have toy till and coins so he can learn currency and adding subtracting amounts, do jigsaws etc)
You may of course do this anyway and it may be enough.
If you do think you'd like to move school, take it slowly. Look around at other schools. Find out about whether they are over or under subscribed in his year. Is there somewhere you'd like him to go which wouldn't involve you moving house? I think you have most of key stage 1 to decide about this, particularly if you are doing the enrichment activities described above.
If you do think you'd like to go for grammar schools, take time to do research. Look at their admission criteria. Do they prioritise certain postcodes or distance from the school or are places just offered to the top so many passes at 11+
Very few, if any state schools do specific preparation for 11+ and the children do need to be ahead ( eg 11+ often covers all of year 6 maths content, even though the exams are taken at the start of year 6 and verbal/ non verbal reasoning are not types of test they will normally meet so practice is advisable. I'd be thinking of some general enhancement tutoring from year 4 (consolidating maths, making sure foundations are solid etc) and specific 11+ tutoring in year 5. You will probably need to do at least the year 5 tutoring whatever school he is in for the reasons I have outlined.