A shewa naʿ generally reflects a reduced vowel. The words you cite have a shewa naʿ where no originally short vowel existed. For example, we find מִקְדָּשׁ but in Shemot 15:17 we have מִקְּדָֿשׁ. We also get an unexpected dagesh in הַצְּפִֿינוֹ (as you note, in Shemot 2:3) and מַמְּגֻֿרוֹת (Joel 1:17).
Blau believes that this dagesh indicates that the shewa is naʿ, and not gemination. He does not suggest a reason for the appearance of the anomalous dagesh.
He states the phenomenon "is quite rare when the letter following the šwa is not a bgdkpt letter, e.g., יִקְּהַת ‘the obedience(?) of’, pronounced yiqəhat Gen 49:1" (Blau, Phonology & Morphology of Biblical Hebrew, p. 114).
I am not convinced that the letter with the dagesh is not geminated; at least, not at the time of the Masoretes. The University of Michigan manuscript, for example, has the form מִקֲּדָשׁ (and not מִקֲֿדָשׁ) (Shemot 15:17), which apparently geminates the qof. As Double AA commented, a hatef would seem to be the natural choice to make explicit that a shewa is naʿ without gemination.
Khan appears to agree. He writes:
Sometimes the preceding consonant is geminated, e.g., מִקְּדָ֕שׁ [miqqaˈðɔː∫] ‘sanctuary’ (Exod. 15.17 < מִקְדָּשׁ [miqdɔː∫]), עִקְּב֥וֹת [ʕiqqaˈvoːχ] ‘footprints of’ (Psa. 89.52 < עִקְבוֹת [ʕiqvoːχ]), מַמְּרֹרִֽים [mammaʀoːˈʀiːm] ‘bitterness’ (Job 9.18 < מַמְרֹרִים [mammaʀoːʀiːm) (Epenthesis: Biblical Hebrew, Encyclopedia of Hebrew Language and Linguistics)