Battery choice: Effect of "pulse current" and "continuous current (maximum)" rating differences
I finally got a moment to check some datasheets and write. I figured someone else would cover this by now, too. But no. So I'll start by pointing out the more important bits of the datasheets to examine. I'm referencing these: Energizer CR2430 and Energizer CR2450:



The basic things to verify are the chemistry and the voltage (as an aside, the chemistry usually determines the single-cell voltage.) Beyond that, you can verify that their operating temperature ranges are the same (and so they are.)

Once that's done, just look at the "Typical Capacity" shown for the two parts. As you already pointed out, the CR2450 has more capacity than does the CR2430. At first blush, this alone suggests that the CR2450 will run longer. Just as you'd hoped.

But there's a second thing to check. Note that the CR2430 says that this capacity is "rated at \$15\:\text{k}\Omega\$ at \$21^\circ\$C" while the CR2450 says that this capacity is "rated at \$7.5\:\text{k}\Omega\$ at \$21^\circ\$C"! The CR2450 rating is for twice the load! This not only confirms the fact that you really can actually expect the CR2450 to run longer, but that the capacity figure might even be a little bit better, loaded more lightly as the CR2430 specification indicates for its capacity rating.

That said, your own device (being a transmitter) is likely using more than the specified continuous drains (CR2430 at \$190\:\mu\text{A}\$ and the CR2450 at \$390\:\mu\text{A}\$.) You also don't know exactly the minimum operating voltage for your device and that affects how long the battery is still useful. So it's possible that you won't get the difference in duration you might expect, comparing the typical capacity figures.

Regardless, it's fairly certain that if you can get around the differences in thickness, you will be okay and you will almost certainly get more duration out of the CR2450.