you probably can as a f2p because you have only 1 crew skill which could be a gathering once and you haven't done flashpoints since obviously for a f2p player they are limited, he's probably in full greens and without relics.
Just a quick note: F2P are not limited ot the "number" of Flashpoints they can run. They can run Flashpoints just as much as a subscriber. They are limited in 2 ways:
1) They can only roll on loot in a FlashPoint 3 times per week. (In other words: if they "pass" on loot (close the loot window without rolling) they don't consume a loot roll; if they press "greed" or "need" they consume one of their loot rolls (win or lose).)
2) They cannot equip artifact (purple) gear.
In light of that, even as a F2P player you can be wearing custom (orange) gear with prototype (blue) mods. You can also equip prototype (blue) relics.
The thing is though that the level 50 and Hard mode FPs reward mostly artifact gear, so unless they purchase the "artifact gear unlock" they will never be able to wear any of the gear at end-game (or any gear of artifact quality).
As such, they have no business being in HM LI because:
1) By default HM Lost Island is not selected in the Group Finder as it is a Tier 2 HM.
2) The Tier 2 HM section in Group Finder clearly indicates that it requires an average rating of 136 gear to be done.
3) While they can be geared for the Tier 1 HM Flashpoints (orange gear with rating 124 mods from the Daily heroics, and rating 126 mods from the Daily Commendations) they cannot equip the necessary gear to be able to reach the 136 rating required for Tier 2 HMs.
4) They cannot equip any of the gear that drops in HM LI becasue it is all artifact quality.4
(Hmmm... that turned out to be more than a "quick note"...