Cuando lo vi ganar la lotería, empezó a dar bote de la emoción.

English Translation

When I saw him win the lottery, he started jumping for joy.

Would “dar botes” work, too?

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The singular is never used here so ‘‘dar botes’’ is certainly the correct option.
Apart from that, after a quick search I couldn’t find any example with botes only with saltos.

https://dle.rae.es/salto#:~:text=dar%20alguien%20saltos%20de%20alegría%2C%20o%20de%20contento

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Wiktionary does have it in the singular:

dar bote

1 to bounce
2 idiomatic) to dabble

(dar bote - Wiktionary, the free dictionary)