It actually looks like she wasn't resetted to level 1 influence but was rather locked at level 1 which is simultaneously max level influence for her (companion tab shows 1/1 influence for her as opposed to 1/50 on other inluence level 1 companions). I think this is the reason she is unable to receive any gifts because there is no levels above 1 for her currently.

I have following bugs that occured with launch of 5.10 update: Character: Dark Side Sith Jaggernaut lvl 70 2V-R8, Broonmark and Lieutenant Pierce: listed in "Unavailable Companions" tab even though I recruited all of them over the course of previous updates. They simultaneously are available in "Other followers" tab and I'm able to summon them (though Pierce is listed as Major there). Jaessa Willsam: Disappeared from "Unavailable Companions" tab with start of 5.10 update, her gear was returned to my inventory automatically as well without any intervention from my side. After completing Ossus story I received her mission notification and recruited her while resuming romance option (my char was married to her while she was available). Now she is listed in new separate tab "Companions: Sith Warrior" while being as rank influence 1 (she was 44 influence rank previously). Also her portrait in companion tab is missing and influence is shown as 1/1 (as opposed to 1/50 for other rank 1 companions) Influence numbers are missing in companion tab as well (for example 1/400 influence on other rank 1 companions). She is unable to receive any gifts to increase influence and her combat stats are similar to rank 1 companion.