I'm confused, is invasion a type of expansion? or a state that comes after failed investment? Any details assuming a faction 'invades' a system with 7 factions in it, what their starting influence maybe when they enter & pend? Do they pend upon entrance or enter in War? I've never seen a faction expand into a system after 7 facitons are present in it.
It's a type of expansion which occurs (rarely!) to stop stagnation in sparsely settled but densely contested regions, and can occur without Investment (and is more likely without Investment). They're rare but not unheard of in the Colonia region ... and extremely rare outside it.
The faction entering gets normal "has expanded" influence levels.
The War (usually) goes active immediately on entrance, no pending time.
Here's a recent example from Chrysus -
https://cdb.sotl.org.uk/systems/44/history
On 15 March, the Nomads of Nevermore find this system is their sole expansion target in range not requiring Investment - the catch is, it's already got 7 factions in it.
As they are not currently pending or active in any conflicts, they can invade, so they appear in the system at 8.4% influence [1] and start a war with the nearest other faction not already in a conflict - in this case, Colonia Defence Force
On 18 March, the War has reached its minimum length, and they have a winning margin, so they win the war
... and on 19 March, Colonia Defence Force are gone from the system reducing it back to a standard 7 factions.
[1] In this case, an exact match with their opponent - but that's not guaranteed, and these wars can start with influence levels below the normal 7% minimum, or with a >10% gap between factions.