Ok - I am going through exactly the same thing as you - lobbying SMT to spend not insignificant amounts of money upgrading an infrustructure that has been underinvested in for years (yet has remained reliable down to my efforts).
The main difference to me and you - is that I have decided to keep CC3 - as it is already paid for, and therefore I am going to spend the next 2-3 years maximising our investment in what is a secure, reliable, and mostly fast CC3 system, rather than ripping it all out and starting from scratch.
How are you spending 10k a year on RM support? Here we are spending about £5k a year on RM support contracts for the network, terminal servers, and smarcache. I am not including server warranty costs in this, as whatever network you have you will still need to spend on extending your server warrantys. Personally I think it's £5k a year well spent as I can escalate any severe problems I can't solve myself, and if me and my tech get run over by a bus, the school can call on RM support for help.
As for your £55k figure - we need to know exactly what that entails in order to be of any help. If it is pureley for hardware and licencing cost, and you are doing the upgrade yourself then I would suggest you could save significant amounts of money on that - if however it is for consultants / another company to come in an install /set up your new network then £55k could be the price you need to pay.
Your right to question your logic and decisions - I question mine when I handed in a plan to spend £20k on infrastructure this year. But at the end of the day my upgrade will provide a faster, more reliable and far more fault tolerant CC3 network, and if this is what the school wants - good on them, of this is not what they want, and they want to spend the 20k on interactive boards - then that is up to them.
Justifying your plan as best you can is all you can do - if they choose to ignore your reccomendations then it will come back to bite them in the rear at some stage. Luckily for us - we had a severe server faliure in 2007 which cost alot to put right and entailed over a week of total downtime - so the school are aware of the effect poor infrastructre can have on teching and learning.
I agree with most of the points in here - especially the ones about rolling plans. I think replacing all of this in one year is unrealistic. There has to be some sort of priority replacement - maybe this will get you some funds this year.
The best way I have of fighting decisions like this is to go with different scenarios to the Bursar/my line manager and outline what has to be done - outline the cost and then present the plan. Plan for Year 1, 2 and 3 - much the same as what has been said on here.
Learn how your line managers work - how they think - are they technical? Once you learn how they think then it makes your job a lot easier. I'm lucky, my line manager trusts my decisions and when I explain to him I don't baffle him with science - just the facts and how it will affect the pupils education.
Sorry, I'm waffling now,
[Edit - oh and yes, I do think it is unreasonable in this climate to expect your school to completely renovate the network - sorry. I understand your reasons for wanting it but with a little bit of spend then maybe things could be improved slightly.]