My question now becomes, how do you transfer the pollen from the anther to the stigma without killing the flower? Maybe I could just cut the flower with tiny scissors and hope that it doesn't dry out too quickly. Also some of the flowers I tried to cut just now fell off; hopefully this won't happen with newer flowers.

I would suggest cutting a slit down the corolla tube that you are going to use as the female parent. You could just cut the entire flower when collecting pollen and not try pollinating the flowers you use as male parents. That would waste half the flowers but might still leave you with enough. Or else you can bag the flower after pollinating it.

Apparently lantana preferentially cross-pollinates, but bagging blooms both before they open and after pollinating them can help make certain the seeds were produced by the cross you made rather than also contaminated by insect pollinators. You would probably need to 'bag' the entire flower head (inflorescence) since the flowers are so small. A small paper bag or a light-coloured (and by weight) small cloth bag would do.

The flowers that fell off were too old for pollinating.

Lanatana's open their flowers in a particular order called centripetally, with the older flowers toward the outside of the flower head (inflorescence) and the youngest flowers towards its centre.