If the current sch only is named on the statement & stmt doesn't name the other nearer school and say current sch is named as parental preference then the LA must carry on paying transport unless or until the stmt is changed
See Dudley MBC case 's.324(5)(a)(i) requires a LA to arrange that the special educational provision specified in a Statement, necessarily including the specification of a school or schools or a type of school to be attended, is made for the child. Thus, if only one school is specified in Part 4 of a Statement, the LA must make arrangements for the child to attend that school'
You can download case here
As you have moved house and circs changed in theory the LA could change the statement and name the nearer school and the current one as parental pref
you would then have a right of appeal to SEN Tribunal
The Tribunal has decided that while it can't usually get involved in sch transport issues it can if the placement is brought into play - which it would be if the LA changed the name of the school on the stmt
So you could argue at tribunal your child needs to carry on attending current sch for e.g. for continuity, specific expertise etc Tribunals are usually quite sympathetic to continuity arguments unless the cost is significantly more
Would your child need transport even if went to the nearer school? If so the Council must provide transport costings for both schools and can only really refuse to pay your choice if the extra cost would be such an inefficient use of resources as to outweigh parental pref (Dudley again)
So yes in theory the LA can change the stmt - name the nearer school and make you appeal - but in practice the hassle and cost of defending an appeal may make LA back down
What LA is trying to do is pull the wool over your eyes and make you think it can automatically refuse to pay now you have moved and another school is nearer - it can't - it has to change the stmt etc etc
Lots of councils seem not to know about the Dudley case (and it was quite controversial but it is the current law so they are stuck with it)
Local Govt Ombudsman can sometimes help (if you haven't appealed yet) see here - you would need to use council complaint process first