Seeing a statistically significant increase in R2 between two models is equivalent to seeing a statistically significant set of predictors in a model that cause that increase. Yet, at its core, the R2 is about prediction. It doesn't sound like you're doing prediction. It's important to be consistent about your goals in an analysis: if you were after prediction, I might criticize your model for lacking certain important predictors, or I might question whether power is something you should be calculating since power concerns inference.
It sounds like your model is about inference: you have some key variables, and a design to assess their effect on outcome(s). When we fit models for inference, we look at the effects--the coefficient terms--to see if the 1-$\alpha$% CIs include 0 or not, and that is the same as measuring if the p-value is less than $\alpha$.
This type of test is essentially a one sample t-test: Is the 95% CI for $\beta$, the regression coefficient, different from 0? For grants, I have used G*Power, and other software, to calculate power for multiple linear regression using the t-test power calculator. You need only specify the mean and the standard deviation of the sample to calculate that power. Those values are identified from previous literature: you need only calculate the half-width of the CI and divide it by the critical value to find the standard error of the mean (for symmetric CIs). If no literature exists, you must make a guess and rationalize it; but again drawing on the literature strengthens your guess.
Lastly, you mention 1 binary regressor which you call an independent variable and 3 outcome variables. I'm assuming further this is a multiple linear regression because you are further adjusting for, say, age, sex, household structure (parental income, education, household size), among a number of other important traits identified in the literature which may be confounding a possible association. If that's the case, further discussion is needed about multiple testing: there are 3 possible hypotheses you are testing. Justify why you should or should not adjust the threshold level for statistical significance to account for an inflated number of false positive findings. This is the last relevant piece to complete a comprehensive power analysis.