Splintered Summons allows you to summon two of one creature/spirit that has a stat block within the spell description.
Battle Familiar clearly states you can only have one familiar.
Which one wins?
Technically speaking, Splintered Summons doesn't work with Battle Familiar at all, because the Battle Familiar is never described as a spirit (neither in the spell nor the statblock), so the spell doesn't summon a spirit. Splintered Summons only works "When you use a spell slot to cast a Conjuration spell that summons a spirit whose stat block is included in the spell description."
But if you want to bypass that, I'd say the Battle Familiar text wins anyway, because its specific to familiars. The only thing I'm aware of that overrides Battle Familiar (and regular Find Familiar) 'one familiar only' clause is Flock of Familiars, which specifically overrides that.
Now, Find Familiar does describe the familiar summoned as being a spirit, but again, the 'one familiar at a time' text should prevent splintered summons from working.
Splintered Summons allows you to summon two of one creature/spirit that has a stat block within the spell description.
Battle Familiar clearly states you can only have one familiar.
Which one wins?
Technically speaking, Splintered Summons doesn't work with Battle Familiar at all, because the Battle Familiar is never described as a spirit (neither in the spell nor the statblock), so the spell doesn't summon a spirit. Splintered Summons only works "When you use a spell slot to cast a Conjuration spell that summons a spirit whose stat block is included in the spell description."
But if you want to bypass that, I'd say the Battle Familiar text wins anyway, because its specific to familiars. The only thing I'm aware of that overrides Battle Familiar (and regular Find Familiar) 'one familiar only' clause is Flock of Familiars, which specifically overrides that.
Now, Find Familiar does describe the familiar summoned as being a spirit, but again, the 'one familiar at a time' text should prevent splintered summons from working.