My question is, though, whether there is any reason to assume that the FE model's error term might not be heteroskedastic.
My interpretation is that what you really want to know is whether heteroscedasticity in the pooled OLS regression implies heteroscedasticity in the FE regression. To that the answer is no. In other words, you cannot test on the pooled OLS regression and conclude that the result also holds for the FE regression.
The model underlying the FE-estimator in its simplest form can be written as $$y_{i,t}=x_{i,t}\beta +\alpha_i + u_{i,t},$$ where we now for simplicity assume $u_{i,t}$ is iid. If you fit a model given by $$y_{i,t}=x_{i,t}\beta +e_{i,t}$$ using pooled OLS and data is generated by the fixed effects model, you have in effect set $e_{i,t} = \alpha_i + u_{i,t}$. Decompose the variance of the error term in the pooled OLS model to get: $$\operatorname{Var}(e_{i,t})=\operatorname{Cov}(\alpha_i + u_{i,t},\alpha_i + u_{i,t})=\operatorname{Var}(\alpha_i)+\operatorname{Var}(u_{i,t})+2\operatorname{Cov}(\alpha_i,u_{i,t}).$$ From this equation it is quite clear that while $u_{i,t}$ is of constant variance (it is even iid), $e_{i,t}$ can very well have non-constant variance. Therefore, evidence of an heteroscedastic error term in the pooled OLS regression is in general not evidence of an heteroscedastic error term in the fixed effects regression.
The null hypothesis in Stata's xttest0 command is that the variance of the unobserved fixed effects is zero, i.e. var(u) = 0. Rejecting this hypothesis means that pooled OLS might not be the appropriate model given that it assumes an error structure $\sigma^2 I_{NT}$. If you have unbalanced panel data you can perform the Breusch-Pagan LM test with the xttest1 command. For a description and more information on this command type
net install sg164_1.pkg
help xttest1
If you want to test whether you should use fixed effects or random effects, you will have to check this with the Hausman test. You can perform this test with the hausman command or, if you have used robust or cluster robust errors, with xtoverid. For the latter command see the help file which includes at the bottom an example of how to test fixed versus random effects.
Best Answer
In a panel model, you have $$y_{it} = \alpha + \beta x_{it} + u_{i} + \varepsilon_{it},$$ so you have two components in your error.
The BP test's null is that the variance of the random effect is zero: $Var[u_i]=0$. Effectively, this would mean that everybody has the same intercept $\tilde \alpha = \alpha + v$, and you can run a pooled regression. I have never been able to reject this null on non-simulated panel data, and you have found that as well. I might consider using het-robust errors and testing again.
To distinguish between RE and FE, you will want to do a Hausman test. The BP test does not help with this decision.