In what approximations, if any, can we assume a constant temperature in a Bomb calorimeter?
You have more or less answered your own question. Enthalpy is a function of state, so you can write an expression of the form $\Delta {H} = {H_f}-{H_i}$, and we know that this depends only on the initial and final states, and not the path taken between them.

You also have the relation, $H = U + PV$, using this and the previous equation, and the fact that $U$ is also a state function (so we can write $\Delta {U} = {U_f}-{U_i}$) 

$\Delta {H} = \Delta {U} + P_fV_f - P_iV_i \tag{1}$

One comment, in the link that you posted equation (6) namely, 

$\Delta {H} = \Delta {U} + \Delta(PV) = \Delta {U} + P\Delta V + V\Delta P $ seems absurd, for finite, non-infinitesimal differences.

For infinitesimal changes, i.e if we write the differential

$\mathrm{d}H = \mathrm{d}U + \mathrm{d}(PV) = \mathrm{d}U + P\mathrm{d}V + V\mathrm{d}P $ this holds true; (chain rule from mathematics)

Anyway, for solids in liquids, the $PV$ term can be neglected on account of their small molar volume. I guess this would be one approximation/assumption you are interested in.

Second, assumption/approximation would be that a gas behaves ideally, and thus follows the equation of state: $PV = nRT$

Plugging it into (1);

$\Delta {H} = \Delta {U} + P_fV_f - P_iV_i = \Delta {U} + R(n_fT_f-n_iT_i) $

This expression is true for finite differences. Now, the finally approximation is indeed $T_f \approx T_i \equiv T$, i.e $\delta T$ is "sufficiently" small, then indeed we can write 

$\Delta {H} = \Delta {U}+ RT\Delta n $

Note however, that we don't necessarily require $\Delta T \to 0$ in a strict, mathematical sense. What can be considered "sufficiently" small depends on your system, your problem, and the kind of solution you need.
What I mean to say is in some cases/applications $-4.96\ \mathrm{kJ}$ is just as good as $-4.88\ \mathrm{kJ}$

There are many other simplifying assumptions in calorimetry when it is taught such as heat-capacities are independent of temperature for example; this only, really true over narrow ranges of temperature and in general, heat capacity is in fact a function of temperature.