I think Jessica is best placed to help with this but I have found some links online that may be of use to you- linking them
here and
here. Wishing you all the best- please do let me know how you get on! 🤞
For these visas, the firm will need a specific licence. As the firm have said they can’t sponsor you, it may be the case they do not have this licence. However, it might be the case they do and are just unwilling to cover the costs of the visa for such a short scheme (the costs will be higher than the costs of employing).
The youth mobility visa is most likely to be your best bet if you can secure this visa your home countries ballot system and can obviously cover the costs/financial commitments.
You could speak to the firm as to whether there are any additional costs they can cover, as once you include the visas, flights and accommodation in London, this is potentially not going to be a cheap few weeks even if you are paid.