To determine the percentage composition of carbon and hydrogen in the organic compound, we need to calculate the amounts of carbon in CO$_2$ and hydrogen in H$_2$O produced from the complete combustion of the compound.

First, let's calculate the amount of carbon in CO$_2$:

The molar mass of CO$_2$ is 44 g/mol, and the molar mass of carbon (C) is 12 g/mol. Given that 0.2 g of CO$_2$ is produced, we can use the following proportion to find the mass of carbon:

$$\frac{12 \text{ g (C)}}{44 \text{ g (CO}_2\text{)}} = \frac{x}{0.2 \text{ g (CO}_2\text{)}}$$

Solving for $x$ (mass of carbon):

$$x = \frac{12 \times 0.2}{44} = \frac{2.4}{44} = 0.0545 \text{ g}$$

Next, let's calculate the amount of hydrogen in H$_2$O:

The molar mass of H$_2$O is 18 g/mol, and the molar mass of hydrogen (H) in a single H$_2$O molecule is 2 g/mol. Given that 0.1 g of H$_2$O is produced, we can use the following proportion to find the mass of hydrogen:

$$\frac{2 \text{ g (H)}}{18 \text{ g (H}_2\text{O})} = \frac{y}{0.1 \text{ g (H}_2\text{O})}$$

Solving for $y$ (mass of hydrogen):

$$y = \frac{2 \times 0.1}{18} = \frac{0.2}{18} = 0.0111 \text{ g}$$

Now, we can calculate the percentage composition of carbon and hydrogen in the organic compound, which has a total mass of 0.3 g.

Percentage of carbon:

$$\% \text{C} = \left( \frac{0.0545 \text{ g}}{0.3 \text{ g}} \right) \times 100 = 18.18\%$$

Percentage of hydrogen:

$$\% \text{H} = \left( \frac{0.0111 \text{ g}}{0.3 \text{ g}} \right) \times 100 = 3.70\%$$

Therefore, the correct answer is:

Option B: 18.18% and 3.70%