You can. I was also not a fan of that system because of the consumables associated with it until I found out you that we could use our own filter paper and membrane. We use PVDF membranes and a extra-thick blotting paper from biorad (Cat#s 162-0177 and 1703960 respectively) and it works very fine. The only thing is that you will have to keep the gel in the transfer/blotting buffer for at least 10mins before you do the transfer. Also, you may have to use a longer transfer time (and voltage, if necessary) of at least 30mins. There's always some protein left on the gel as I can tell, especially the high molecular weight ones.
There is now a solution for the TBT that will deliver the same speed but at a lower price it can be found here http://www.bio-rad.com/en-us/sku/170-4275-trans-blot-turbo-rta-midi-lf-pvdf-transfer-kit-for-40-blots?parentCategoryGUID=LGOQBW15
Our lab recently bought this system and we could not achieve good results. I would like to know what kind of transfer buffer do you use during transbloting?
In the "Alternatives to Trans-Blot Turbo PVDF packs" discussion I have placed the information about our protocol.
"Hi! There have been so many people interested that I decided to place our protocol in a repository for Protocols. You can find it here: https://www.protocols.io/view/trans-blot-turbo-transfer-with-home-made-buffers-ghhbt36
Please reference this DOI and the original papers. That way we support the repository and as well those who worked so hard to find the things we were able to adapt!
Have fun!"
Alternatives to Trans-Blot Turbo PVDF packs - ResearchGate. Available from: https://www.researchgate.net/post/Alternatives_to_Trans-Blot_Turbo_PVDF_packs/4 [accessed Jan 25, 2017].