I'm going on the assumption that, when you say "transformer", you mean a thing that you plug into the wall (AC power) and get 12V (DC power) out the other end. If so, the more accurate term would be "power supply". The term "transformer" specifically refers to the device that uses windings of wire around a core to convert one AC voltage to a different AC voltage. More stuff is required to get DC power out of it.
Based solely on the 50W solid + 50W flashing numbers, you'd need a supply that could handle the peak usage, 100W, when both lights are on. 100W / 12V = 8.3A.
However, there will be a relay causing the second bulb to flash, which will eat a bit more power. My guess is that it'd be safe to assume it's under 0.5A, but I would personally test the assembly with a multimeter while it's powered. Unfortunately, that may not be possible if the flash is fast and/or the multimeter is slow. The reading might not stabilize by the time the relay switches the second bulb on or off.
Skipping the manual power measurements, the whole thing is likely under 9A, but I wouldn't run it off a 9A power supply for two reasons:
(1) It's not a good idea to continuously run a power supply at its maximum rated output.
(2) There's a pretty good chance that the manuafacturer of the power supply was a bit generous in their rating. If you measure the voltage across the power leads and it dips significantly when both bulbs are on, then you're pushing the power supply too hard.
Using the assembly as-is, I'd probably use a power supply rated at 15A, at least. After a very quick look, they seem to be around CA$50 on Amazon and CA$30 on AliExpress.
I would most likely replace at least the constanly-on bulb with an LED to reduce the power consumption and heat generation (every Watt of inefficiency in an incandescent bulb is released as heat)*. If an LED wouldn't spoil the ambiance for your friend, the how-to would warrant another question, if it hasn't aready been asked.
Good luck!
*Edit: More accurately, every Watt of inefficiency in anything is released as heat. The reason LEDs are more efficient than incandescent bulbs is because less heat is released per quantity of light produced. And, ultimately, that light gets absorbed into some object and it too gets converted to heat. And friction -> heat. Etc., etc. And that's the depressing reality of entropy. Y'know what cheers me up? Pretty lights!