I would just do a regular bootstrap test:

• compute the t-statistic in your data and store it
• change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group 1 and add the overall mean, and do the same for group 2, that way the means in both group will be the overall mean.
• Take bootstrap samples from this dataset, probably in the order of 20,000.
• compute the t-statistic in each of these bootstrap samples. The distribution of these t-statistics is the bootstrap estimate of the sampling distribution of the t-statistic in your skewed data if the null-hypothesis is true.
• The proportion of bootstrap t-statistics that is larger than or equal to your observed t-statistic is your estimate of the $p$-value. You can do a bit better by looking at $($the number of bootstrap t-statistics that are larger than or equal to the observed t-statistic $+1)$ divided by $($the number of bootstrap samples $+1)$. However, the difference is going to be small when the number of bootstrap samples is large.

You can read more on that in:

Chapter 4 of A.C. Davison and D.V. Hinkley (1997) Bootstrap Methods and their Application. Cambridge: Cambridge University Press.

Chapter 16 of Bradley Efron and Robert J. Tibshirani (1993) An Introduction to the Bootstrap. Boca Raton: Chapman & Hall/CRC.

I would just do a regular bootstrap test:

• compute the t-statistic in your data and store it
• change the data such that the null-hypothesis is true. In this case, subtract the mean in group 1 for group 1 and add the overall mean, and do the same for group 2, that way the means in both group will be the overall mean.
• Take bootstrap samples from this dataset, probably in the order of 20,000.
• compute the t-statistic in each of these bootstrap samples. The distribution of these t-statistics is the bootstrap estimate of the sampling distribution of the t-statistic in your skewed data if the null-hypothesis is true.
• The proportion of bootstrap t-statistics that is larger than or equal to your observed t-statistic is your estimate of the $p$-value. You can do a bit better by looking at $($the number of bootstrap t-statistics that are larger than or equal to the observed t-statistic $+1)$ divided by $($the number of bootstrap samples $+1)$. However, the difference is going to be small when the number of bootstrap samples is large.

You can read more on that in:

• Chapter 4 of A.C. Davison and D.V. Hinkley (1997) Bootstrap Methods and their Application. Cambridge: Cambridge University Press.

• Chapter 16 of Bradley Efron and Robert J. Tibshirani (1993) An Introduction to the Bootstrap. Boca Raton: Chapman & Hall/CRC.

• Wikipedia entry on bootstrap hypothesis testing.