Certainly possible. I have done a JAA IR in my own plane, in the UK.
However many (possibly most) FTOs will not handle a customer aircraft. I guess that in some cases they can't be bothered with putting it on their paperwork (though it does seem a trivial exercise) and in other cases they don't want to handle a lower margin customer. I have been told by one FTO that they don't like doing it because too many such customers want something for nothing.
Last year I researched the options for this around Europe and incidentally found that UK seems to be the only place for doing this in an N-reg aircraft. In the rest of JAA-land it has to be EASA-reg, but the % of non-UK FTOs willing to do customer aircraft appears to be miniscule.
You need the insurance clause added (trivial), and Whopity is right re the £177.
The HUGE plus of using your own plane is obviously currency on type, and the training is of more real benefit to you. Also the marginal operating cost is likely to be much lower than that of an FTO one.
Unlike the IR, the CPL can also be done outside JAA-land, it appears, FWIW.