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In his sefer Teshuvos V’hanhagos, Rav Moshe Shternbuch brings down this Reb Chaim (he actually says it b’shem the Bais Halevi). Ironically he quotes the gemara in Rosh Hashana as a raya to this svara although he does not explain how. ??? ????? ???? cannot explain how it is a raya, but I will try to explain how it is not a kashya according to Rav Shlomo Miller’s pshat. Parenthetically, I don’t know if Rav Shternbuch agrees with Rav Shlomo Miller’s pshat; it’s very likely that he doesn’t.
There are three types of Safek D’oraisa (that I can think of):
1)A safek whether or not you have already performed a mitzva – in such acase it is pashut that you do it again, thereby resolving the safek.(E.g. you don’t remember if you put on tefillin)
2)A safek whether by doing a certain action you will be able to perform a Mitzva D’oraisa. The case in Rosh Hashana would fall under this category.
3)A safek whether a certain action is a fulfillment of a Mitzva D’oraisa. Both The Ran’s case and techeiles fall into this category.
Rav Shlomo Miller’s pshat is that we apply Safek D’oraisa L’chumra any time where it will end the situation. Therefore in category one we for sure apply it. In ctegories two and three it depends on the individual case. In the Ran’s case, there is a safek whether there is a chiyuv D’oraisa to take Arba Minim. Once you take them, the situation is over – it was either the first day of Succos and there was a chiyuv, or it was the second day of succos and there was no chiyuv. Either way there is nothing more to be done. Similarly in the case in Rosh Hashana once you travel to the shofat city the situation is over – either they will be able to be motzi you orthey won’t. Either way there is nothing left to do. Therefore in both of those situations we would apply the rule of Safek D’oraisa L’chumra.
However, when it comes to techeiles, even after donning your murex trunculus tzitzis there is still a safek whether or not you are fulfilling the mitzva of techeiles. And unlike the other two cases the siyuation is not over because you can still make “techeiles” with any number of other potentially viable creatures(e.g. cuttlefish, janthena etc.) In such a case we would not apply the rule of Safek D’oraisa L’chumra.
The only problem that I can think of with this pshat is that I don’t know of any other case where this svara would apply. However that is not such a dochek because accornding to Rav Shternbuch, the Bais Halevi specifically said this svara in response to the Radzyner so it is very possible that this is the only case where the svara is relevant.
On a side note, it would seem according to this pshat that Lu Yetzuyur there were no other snails/fish in the world then we would apply the principle of Safek D’oraisa L’chumra and use murex trunculus techeiles.