We're already a hotshot at Alchemy, so we may not actually want to invest in that heavily. If we spend 7000, I say...
3 counts of Summoning, bumping that skill to 91.
4 counts of Enchantment, bumping that skill to 96.
I'm sure there will be ways to bring Alchemy up to speed, but both Enchantment and Summoning seem to see good use in battle situations now that we're no less Warrior and more Historian.
THIS DOES NOT MATH*
It's been repeatedly emphasized that we need (for best results) multiple schools at 100%, and it certainly sounds like this is our very last chance to train magic. I'm baffled as to why we would take the strategy of getting multiple schools to just below 100% - if we run into a bunch of 100 skillchecks with no opportunity to raise our skills higher first...
* You appear to have assumed that Enchantment and Summoning are both at 'high' skill, and only receive 10% each time. While we don't know where high skill begins, we do know that alchemy is in that range, and enchantment and summoning are not (they're medium skill, and thus would gain 15 until they reach high skill).
A reasonable guess would be that high skill starts at 75 or 80.
We need 2000 gold to max out alchemy.
If high starts above 71, then we will need 4000 gold to max out enchantment. (+15=71, +15=86, +10=96, +10=100)
If high is at 75, then we will need 4000 gold to max out summoning. (+15=76, +10=86, +10=96, +10=100)
If high is above 76, then we will need 3000 gold to max out summoning. (+15=76, +15=91, +10=100)
We would apparently need 9000 or 10000 gold to max out all three, which would leave insufficient gold for hiring an advisor.
With ASE and high barrier at 80: A100, S100, E86.
With ASE and high barrier at 75: A100, S100, E71.
With AES and high barrier above 71: A100, S76, E100.
With 0A, 3S, 4E, and high barrier at 75: A87, S96, E100
With 0A, 3S, 4E, and high barrier at 80: A87, S100, E100 (which is numerically better than ASE, but if the high barrier isn't above 76, then we only have one maxed-out skill because summoning only reaches 96)
So I would go with
Alchemy, Summoning, Enchantment, or
Alchemy, Enchantment, Summoning. It might be possible to raise both enchantment and summoning and see at which point they reach high skill, and then decide which one to prioritize based on which one will give the highest third skill. Skipping alchemy seems dangerous because we could end up with only one additional maxed out skill instead of two, though.