If you bypass the switched power lead by using a direct connection to the battery, the radio will remain ON when the key is removed which is not good. There will always be that one time you turn the volume all the way down and forget to turn off the radio and then the car won't start after sitting for hours.
Adding a large capacitor in line with the switched power lead will probably do the trick, but it will cause the radio to play for several seconds after you remove the key until it completely discharges.
Either way, you will need to know which wire is the switched power lead for the radio.
There is another way in which to have your radio stay on while cranking. At your ignition harness, the are a number of different wires for different purposes.
12v Constant: it's a constant 12v line
Started: it gets 12v only when you are cranking the starter
Ignition: it gets 12v while the key is on, and while cranking
Accessory 1: it gets 12v while key is on, shuts off while cranking
Accessory 2: same as Accessory 1.
A typical radio is hooked to an Accessory lead... So, what you need to do is cut the Radio Switched 12v lead, securly tape off the car side of the wire so it cannot touch ground or move around. Take the radio side and extend it down to your key cylinder. The Ignition wire is Black with a Yellow stripe. Get an inline fuse (10a should be fine) and use it to connect the Radio 12v Switched wire (Radio Side) to the Ignition wire. Fuse should be located as close to the thick ignition wire as possible. Now, your radio should not turn off during cranking... and you shouldn't have to put a CAP behind the radio.
I hope this helps... I know the post is old, but I figure somebody might want the info.