The short answer is: you do need the "main", i.e. non-time varying effect of the covariate.
Based on the help file for
stccreg, if $X$ is declared a time-dependent covariate, then the time-dependence is modeled parametrically as $X(t) = X\cdot f(t)$, where the default is $f(t)=t$. That means that at time 0, the time-dependent portion has no effect, and then the effect increases/decreases linearly in time (on the log-hazard scale). So in this case, omitting the $X$ as a time-invariant predictor has the same effect as omitting the intercept from a linear regression model: you are eliminating any immediate ($t\approx 0$) effect of $X$. This is probably unwise unless specifically desired.
Stata will allow you to change the multiplier $f$ function, so the interpretation of the time-invariant part might change, but $f$ will always have to be a predefined function with no unknown parameters.
As for testing, if Stata does not give a test in the default output, you could fit the model with and without the time-varying part, and do a likelihood-ratio test with 1 degree of freedom (assuming $X$ has one degree of freedom).