The "size constrains" I was thinking about the physical dimensions. But I don't believe it matters for you after the explanation. Now 6000 TX/year, it seems that you'll be sending very little data and not very often.
To simplify your calculation, I would say that your usage could be split in three parts:
Part A: entering the command/score in a Keypad. That can be done with sleeps between each keypress, but honestly, it should be much easier to code everything to run at 1MHz and wake-up the transmitter node on the first key press and give some time for the user press the keys.
Part B: would be the transmission itself, where the MCU wake-up the Radio, fills the Radio buffer and "sends" the message. You could even have a local LCD showing the message to be sent/updated on the score board and a button to transmit it.
Part C: all at sleep.
For the "Part A", if you have a look at the ATmega328p datasheet (http://www.atmel.com/Images/Atmel-42735-8-bit-AVR-Microcontroller-ATmega328-328P_datasheet.pdf), on page 379, you'll see that the MCU consumes less than 1mA (0.6mA actually) when running at 1MHz. For a keypad only and one or two LEDs, that's more than enough!
Now, for the radio - "Part B" - I haven't measured it, but if you're transmitting only a few bytes, it should take less than 10ms. Don't worry about re-transmission, as it would be just a matter of pressing the "send" button again.
Finally, I'm assuming the score-board will be a RX only node, plugged into "mains", so it'll be always listening.
Here some calculations - you might need to adjust according to your finds/tests:
Part A - MCU Running at 1MHz + some pull-ups or what ever is necessary for your Keypad, buttons, etc - rounding up to 5mA for 3 seconds
Part B - MCU Running and Radio Transmitting - 100mA for 10ms
Part C - Sleeping, 1uA for the MCU without anything running + 20uA for the Regulator + Radio at sleep - lets round it up to 0.03mA (30uA). Now, let's suppose that instead of 16 presses a day, you have 24 presses a day, one per hours, to simplify the maths.
Here the maths for the average consumption... the idea is to bring all numbers down to the same units. In this case Ill use milliseconds (ms) and milliamps (mA). Also, you need to know the duration of your cycle, where after that period everything starts to repeat, in this case it's 1h (3600 seconds or 3600000 milliseconds).
Part A: 5mA * 3000ms = 15000mA*mS
Part B: 100mA * 10ms = 1000mA*ms
Part C: 0.03mA * (3600000ms - 3000ms - 10ms) = 107909.7mA*ms
Average: (15000mAms + 1000mAms + 107909.7mA*ms) / 3600000ms = 0.0344193611111111mA or 0.035mA
Now, considering you have a 9V, which has a 500mA/h capacity, and you're using an LDO to down it to 3.3V, well, you get no more than the 500mA/h as all excess energy will be burned out as heat. Also, a 9V battery will be aroung 4.8-5V by the end of its life... But that still probably good enough for your project, specially as the capacity seems to improve on slow discharge profiles: http://data.energizer.com/PDFs/522.pdf - see first chart.
Taking everything above, the 9V battery would last around: 500mA/h / 0.035mA = 14285h = 595 days = 1.6 years. Considering the battery will self-discard 2-3%/year you might wish to round thing up a bit more and take of 10% of that. Giving you a final figure of 1.4 years.
So, I gave a bit of "fat" on all parts of your project, specially at the sleeping current. At the end, if replacing the battery every year is something acceptable you might even be able to attach a little LCD to display the data locally before sending the message to the score-board!
The pitfall might be some leaks... You just need to make sure there's no current leak on the circuits. Using a Mosfet to power things-up, including the Keypad, battery monitor (voltage divider) and radio might be an alternative to eliminate current leaks.
Make sure you also test the Radio range, pick a slower speed in case you have trouble and follow previous suggestion of adding a nice 10uF ceramic capacitor very close to the RF module. Hope the example above helps you.