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.
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.
Best Answer
Briefly, the Hausmann Test checks the specification of the model. In particular, you can use it to test if you should be using the RE or FE specification. The BP Test is a check of heteroskedasticity (does the variance depend on the independent variables). You can certainly use the BP Test independent of the Hausman test for any linear model. However, if the Hausman test tells you that you should be using RE then the BP test can tell you wether or not there is heteroskedasticity at the individual level. If none is found then you can just use pooled regression. Basically, the BP test is testing if the individual error term has variance zero, if it does then pooled regression will capture it.
Here is a reference for more details: https://www.princeton.edu/~otorres/Panel101.pdf