If its yours (share will do) it doesn't matter it can be a permit aircraft. But check your insurance sometimes they have experence clauses in them for you flying solo and use for training.
It all gets a bit complicated if you borrow a mates plane and i wouldn't like to give you any advice on that. It is done quite frequently but I am sure its in a grey area of the law
The reason i stated I might get shot down in flames is that it may have all changed in the last 3 years and this option might have been removed.
As such there isn't such a thing as a JAA tail wheel conversion. There is only tail wheel differences which is just a sign off by the instructor with a stupid sticker. The CAA doesn't get any paper stating you have done the training, the only evidence you have done the training is the sticker in your log book. Its the same for pressurisation, constant speed props, turbo's and retractable.
I am sure cash plus expenses will get the job done pretty quick.