1. Determine the probability of Joe getting beat up:
Joe has a probability of getting beat up if he dresses as JC Chasez or Justin Timberlake. The probabilities are:
- JC Chasez: $25\%$
- Justin Timberlake: $60\%$

Since there are 5 members of NSYNC and Joe is equally likely to dress as any of them, the probability of Joe dressing as JC Chasez or Justin Timberlake is:
$$\frac{1}{5} \text{ for JC Chasez} + \frac{1}{5} \text{ for Justin Timberlake} = \frac{2}{5}$$

The combined probability of Joe getting beat up is:
$$\left(\frac{1}{5} \times 25\%\right) + \left(\frac{1}{5} \times 60\%\right) = \frac{25}{100} \times \frac{1}{5} + \frac{60}{100} \times \frac{1}{5} = \frac{25}{500} + \frac{60}{500} = \frac{85}{500} = 17\%$$

2. Set up the equation for Kathryn's probability:
Kathryn needs at least a $65\%$ chance to tell Joe how much she likes him. If Joe does not get beat up, Kathryn needs to bribe Max to play Joe's favorite song. The probability of Joe not getting beat up is:
$$1 - 0.17 = 0.83$$

Let $x$ be the amount of money (in dollars) Kathryn gives to Max. The probability of Max playing the song is:
$$\frac{10x}{100} = 0.1x$$

The combined probability of Kathryn getting a chance to tell Joe is:
$$0.17 + 0.83 \times 0.1x \geq 0.65$$

3. Solve the inequality:
$$0.17 + 0.083x \geq 0.65$$
Subtract 0.17 from both sides:
$$0.083x \geq 0.48$$
Divide both sides by 0.083:
$$x \geq \frac{0.48}{0.083} \approx 5.783$$

Since Max only accepts money in increments of $10, we round up to the nearest multiple of 10:
$$x = 60$$

The final answer is $\boxed{60}$