Audiolabyrinth,
If the 5814 will not work in your player, the 12BH7 surely will not. It draws exactly double the heater current that the 12AU7 does.
I seriously doubt that the 5814 will do any harm to your situation. It only draws a fraction more heater current than a normal 12AU7.
To answer, the 10M 12AX7 is far and away my favorite 12AX7. Smokes the Telefunken for sure. They are pricey also though...
Some of the better 5751 tubes will compete sonically, but I have found no better 12AX7 than the 10M Mullard. Very 3D, extended, with deep powerful bass. Also not as overly warm as the typical Mullards.
Another great tube is the 1950's version of the GE Black Plate 5751's. This tube gets damn close to the 10M, but they don't last near as long.
The 7318 is the best 12AU7 I have tried. Beats 12BH7, ECC99, or any other brand or variant you can think of. It is very special, and yes, it is big on tone, just like the 6SN7.
Cheers!