If there are multiple possible approximations, I’m looking for the most basic one.
You can approximate it with the multivariate normal distribution in the same way that binomial distribution is approximated by univariate normal distribution. Check Elements of Distribution Theory and Multinomial Distribution pages 15-16-17.
Let P=(p1,...,pk) be the vector of your probabilities. Then the mean vector of the multivariate normal distribution is np=(np1,np2,...,npk). The covariance matrix is a k×k symmetric matrix. The diagonal elements are actually the variance of Xi‘s; i.e.npi(1−pi), i=1,2…,k. The off-diagonal element in the ith row and jth column is Cov(Xi,Xj)=−npipj, where i is not equal to j.