Nikki, sorry I wasn't around earlier.
The law on flexible working is really complicated. There are two different issues which itersect.
The first is flexible working legislation. This is the law that gives you an express right to request flexible working. As you have discovered, it is quite easy for employers to turn down these requests provided that they follow the proper procedures (letting you appeal, etc). You can sue them if you think that the reasons they gave were not good enough, but the financial remedies are punny and it's basically not worth it.
As a result of the flexible working legislation being a bit toothless, women whose requests are turned down tend to claim indirect sex discrimination. The argument goes like this: women in society are still far more likely to have primary childcare responsibilies than men. Thus, if you make a woman work full time, you are applying a criteria which has a disproportionate effect on women. This is indirect sex discrimination and is illegal unless the organisation can properly justify it.
As you can imagine, this is quite a difficult route to go down. If you tell your employer that you are thinking of suing them for sex discrimination, it can make life rather difficult in the office/work place.
I would personally agree with what your union representative said about going in for the time being but making your position clear.
In terms of taking it further, I think that your union needs to fight your case for you. I think that they need to question two things (not knowing your job):
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Would it really affect customer requirements? BTW, do you agree that the job needs someone there full time;
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The fact that they haven't tried to advertise externally.
If you really wanted to get into this on a legal level, those issues could point to a lack of good faith in trying to find a part time position, which could be a pointer towards sex discrimination. The sad thing is that, even if it's true, you can sue for compensation, but almost nothing you can do can actually make them give you the job part time.
In terms of the alternative employment, there is no actual obligation to look for other jobs. That only applies to reduandancy. However, again, it might feed into the sex discrimination issue.
That's probably the best I can help for the time being. Do use your union. They are a really useful resource and know what they're talking about. Even ignoring the legal position (which I am sure is not as clear cut as you probably hoped), the commercial pressures a union can put on a company (no-one wants to fall out with the union) can have an effect. Get that rep fighting your corner!
Good luck!