If the child has just gone into Y11 then there is a chance to catch up; but the parents would need to get him/her moved into a higher tier set pretty quickly. There is a fair bit of extra content in higher tier - a foundation paper basically has one higher tier level question at the end which is aimed at the better candidates getting a C, which is the highest achievable on the foundation paper. There are so many boards and variations on the modular theme that it is not possible to say which papers any child is sitting - you would need to get chapter and verse from the school. I would suggest, tentatively, that if a school has taught a child at foundation level for the whole of Y10, and left them in that set in Y11, the school doesn't think they would manage higher tier. I would strongly recommend talking to the school; I assume as this is coming from you Tolstoy, that there are very good reasons for thinking the child could achieve a B or above.
I didn't know it was possible to get L5 at KS1, but that would be a very bright child indeed and it would be odd to be doing foundation GCSE. If you meant to say L5 at KS2 - as you know, this is not a great indicator with the teaching to the SATs which has tended to take place. Children's NC levels are often subject to some 'adjustment' over the course of Y7-8, suggesting that high achievement levels in Y6 might have been a tad inflated.
Schools often mix and match. At our local comprehensive, they enter most children for Foundation maths in the November. If the children get the 'C' at that point, they are then entered for the Higher paper in the following summer. If they don't, they do the Foundation as a resit in the summer.
What a great idea!