You will need to use a power transformer that has a secondary voltage less than ~300vac, or use a second 6vac winding.
The 6X4 is rated for 450v maximum between heater and cathode, which will be exceeded by the 320-0-320v secondary of the AA764 Champ, and that assumes wall voltage or a transformer with a 110v primary don't result in more than the rated voltage.
The second 6.3v winding would power only the 6X4 heater, and would be connected to the 6X4 cathode at the socket to minimize the heater-to-cathode voltage.
I'm a bit confused because the indication for the first cap changes from one datasheet to other as an example
I will say I had a Bogen amp that came with 16uf on a rectifier tube that had 10uf specified as it's max.
One day we'll get everyone to see those are not maximum ratings.
So if you do nothing else, the cap values listed might be considered "advisable values" for you to adhere to. They take into account the maximum cathode current rating for the rectifier and the peak charging current of the filter cap.
But power supply impedance can limit cap charging current and allow a bigger filter cap to be used. That impedance is often the resistance of the PT half-secondary plus the resistance of the PT primary, referred to the secondary. That is what
Kagliostro's 40uF 6X4 sheet is showing; "Rt" is that transformer impedance, and is part of the assumption that allows the bigger filter.
Some equipment actually included small resistors in series with each rectifier plate, at the socket, to increase the supply impedance and limit the peak charging current.
There's a mess of charts and tables in RDH4 that can be used to work through all the steps of a power supply design and determine the needed supply impedance for a given rectifier and filter cap, but it's an involved process and few seem to actually make use of them. The charts simplify the mass of calculations you'd have to do to get the same answer.