Is the Chalice really needed? I run mana regen blues, yellows and the mastery and never find mana to be an issue outside of the very early levels, where I simply have to moderate my use of boomerang blade. Considering that while yes, Sivir is a pusher champ, trying to push towers at such a low level is often unwise in the first place (especially if there is an enemy jungler), I can't really see a major benefit to getting it.
By the time I'm at the point where I want to start pushing (somewhere around lvl 8 or so unless I dominated my lane), I don't need any more mana support.
I suppose I can see why you might want to get one, if you laned against say, a Kassadin that was kicking your butt in the lane (though how you'd lose to a low level Kass as Sivir is beyond me...), then maybe the early Chalice would be useful for safer farming and the MR for extra protection against harrass. But other than that...
If you really want to go the Manamune route, Tear of the Goddess is also sufficient. You can swap out either of them late game for superior damage sources.
You can of course, rely exclusively on Spell Shield for early game, but I find that clever opponents will work around that. My last game, Vlad just refused to target me with spells ever. I blocked one of his transfusions after reading his pattern and he just exclusively targeted minions with that spell from then on. Not until we got into Team fights mid and late game did he try to hurt me again.
I've also laned against Ezreals that have either purposefully or accidentally missed their skill shots that would otherwise be free mana for me just to juke my spell shield up. Not only causing me to waste mana, but to put the spell on cooldown while they hopped in for some quick harass. Fortunately, BB is superior to any harass Ezreal has at level 3-5.