the order of lighting menorah and shabbos candles: tadir vs. pirumei nisa

I thank R’ Chaim Markowitz for calling to my attention yesterday (and hopefully I did not steal his thunder) to an interesting machlokes regarding hadlakas menorah tonight. The BH”G (quoted by the Tur) writes, and this is the common practice, to light the menorah first and afterwards shabbos candles, because lighting shabbos candles is effectively accepting shabbos. Tosfos argues and holds that kabbalas shabbos is distinct from the act of lighting shabbos candles. If one holds like Tosfos, is there any reason to insist on lighting Shabbos candles first, or is it a toss-up between both mitzvos with no real preference? Tosfos writes that the shabbos candles are lit first but does not say why, leading achronim to write that b’dieved the order does not matter - it sounds like a toss-up. The Rashba, however, insists that Shabbos candles should be lit first because of the rule that “tadir v’she’aino tadir, tadir kodem”, a mitzvah which is performed more frequently always comes first. It seems to me that perhaps Tosfos deliberately omits this reason – perhaps a mitzvah of pirsumei nisa outweighs a mitzvah which is tadir. Two other cases relevant to Shabbos Chanukah may hinge in part on this question:

1) While many people will light menorah at home and then go to shule for mincha today, the Birkei Yosef holds that mincha should be said before lighting menorah. He explains that in the Bais haMikdash, the korban tamid, which our tefila of mincha corresponds to, preceded the mitzvah of lighting menorah. Without that sevara, one might argue that the mitzvah of tefilah should take precedence, as it is tadir – but then again, perhaps pirsumei nisa comes first! (I have not researched this issue, but it seems to me that if one accepts this Birkei Yosef, which is cited in the Sh’arei Tshuvah as well, there is good room to argue that it is better to daven mincha b’yechudus before hadlakas menorah than to daven b’tzibur after hadlaka).

2) There is a great debate among achronim whether to make havdalah first or light menorah first on Motzei Shabbos. Here the competing factors are delaying declaring Shabbos over, tadir, and pirsumei nisa. One other note with respect to Motzei Shabbos: I believe Ma’aseh Rav quotes GR”A as rushing to end Shabbos immediately at the zman in order to do hadlakas menorah as soon as possible after Shabbos. This perhaps is tied to the GR”As opinion (based on Rambam) that hadlakas menorah should rightfully be done immediately after what we call shkiya. In any event, whether before or after havdalah (depending on your minhag), the hadlakah should be taken care of ASAP.