Here is an answer based on the following interpretation of the SWF : there is no "true" SWF, but SWFs are observable in principe, they simply represent the preferences of the policy decision maker.
Under this interpretation, the policy recommendation are relevant despite depending on the specification of the SWF, precisely because different decision makers have different preferences. So there is no problem with studying different configuration of the SWF and getting different policy recommendations. In fact this is rather desirable as it makes the theory relevant to a larger number of decision makers.
Now, whether you are aiming at policy recommendations for the average decision maker, or whether you are giving policy recommendation to a particular decision maker, figuring out an exact social preferences might be complicated. Data may be insufficient to pin down particular preferences without much uncertainty. And if you ask the decision makers to state their preferences, they might be uncertain about their own preferences.
As is to be expected, the more uncertainty about the decision maker's preferences, the less precise the policy recommendations will be. But there is still a whole lot you can say even under relatively large uncertainty. As a matter of fact, most of the literature on optimal taxation has focused on providing general policy recommendations which hold for a wide class of SWF.
A very good example is Roëll, A. A., 1985. A note on the marginal tax rate in a finite economy.
Journal of Public Economics 28, 267–272's very weak redistribution assumption (VWR), which was informally presented earlier in Guesnerie, R., Seade, J., 1982. Nonlinear pricing in a finite economy. Journal
of Public Economics 17, 157–179. The VWR is a property of the solution of the taxation problem which formalizes the idea idea that the decision maker is egalitarian in at least a very mild way. To give a taste of the weakness of the assumption, notice for instance that
If the utility function U is strictly concave, leisure is a normal good, single-crossing is satisfied, the social welfare function $W$ is differentiable, strictly concave, increasing, and symmetric, and at a solution of the taxation problem, the lower level of income is positive, then VWR is satisfied (Brett and Weymark, forthcoming)
It turns out that the VWR is sufficient to derive many important policy implications. For instance, under the VWR (and single crossing), marginal tax rates must be non-negative at a solution of the taxation problem. Also under the same assumptions, the marginal tax rate of the highest type should be zero, and any type different from the highest type whose consumption and income are strictly positive must face a strictly positive marginal tax rate. Provided the non-normative assumptions of the model hold (e.g. single crossing, but also the fact that decision maker's preferences are representable by a SWF, more on this below), these are policy recommendation you can safely make to any policy decision maker with some form of egalitarian concerns, even if you do not know the exact degree to which they are egalitarian.
Another question you are raising is whether policy decision makers' preference can be represented by a SWF at all. To say the least, I am not a big fan of the cardinal utility framework of SWF. Some decision maker's preferences might be representable through SWF, but I believe most would not. However, there are alternatives with applications to optimal taxation, see for instance the last chapters of A theory of fairness and social welfare, by Fleurbaey and Maniquet.
The approach described in A theory of fairness and social welfare, by Fleurbaey and Maniquet is not only ordinal, it is also purely axiomatic. Decision makers may have a hard time figuring out which SWF represent their preferences. With an axiomatic approach, they simply have to decide on the axioms they like the best to "discover" their preferences. This is often much simpler than trying to figuring and spelling out your full preference relation.
