I think you need to bear in mind that if you do this then you are depriving a child who is genuinely entitled to be there to a place in this school. Perhaps this doesn't concern you.
In which case you need to be aware that this is not going to be as simple as you think. You will need to physically move into the rented home for quite some time. If you keep your old house and do not rent it out the LEA can check council tax records to make a decision about which is your true address (and may well decide that suspiciously renting in catchment just before applications are due makes this a false address even if you are actually living there full time).
They can also ask for which doctor you are registered with to check you are not really still living in your 'old' area.
Most LEAs state that the address on the form must be your child?s permanent place of residence. It should not be a business, relative or child minder?s address. You are not permitted to use a temporarily rented address to secure a school place for your child. It is fraud.
The LEA will investigate any cases where there are any doubts - such as recent moves.
Any application that uses false information may be subject to legal proceedings. If an application is found to be fraudulent after a place has been offered, the place will normally be withdrawn.
The school would be well within it's rights to withdraw the place they have offered your child if you move out of the catchment area even once they have started school. Schools are well aware of parents who try stunts like this to get into a 'good' school - and are on their guard against it.
I don't know who you spoke to at the LEA but it sounds like whoever answered was puzzled and wondering 'why is this woman asking me about making what is clearly a false application?