To my knowledge, it is not possible to route a portion of a payment through a channel. So if you have channels with BTC 0.001 of capacity on each side, you would be unable to handle any transactions above this amount.
it is possible to route a portion of a payment though a channel.
Yen<>dan<>chad
^ ^
v v
zoe<>alice<>bob
if alice wants to pay dan 0.002
its as easy as just gossiping a route through zoe-yen-dan for 0.001
its as easy as just gossiping a route through bob-chad-dan for 0.001
=dan gets his 0.002
dont think of it as a single payment that needs to be split. . but 2 payments that need to be joined
simple solution. dont want to pay 0.002 and request a single invoice for 0.002 and then find a way to split it
request 2 invoices of 0.001 via different channel routes