What is your Intelligence score?
I think we can look at this like a program. We are doing so ONLY as a method of thinking about this problem, NOT because this is how the game works. Below I will describe an update loop and the like, but the game does not run on such a thing. This is based on the phrasing.
The headband is not a one time occurrence, it is a continuous loop. It uses the word while.
Your Intelligence score is 19 while you wear this headband.
If we look at it like a program, we're setting the Intelligence to be 19 every "update". Or rather, every moment in time. Is the headband on? if yes, your intelligence is 19.
Feeblemind is a one time effect, meaning one time and one time only, we're setting the creatures Intelligence and Charisma to be 1. It has a duration of "Instantaneous", which implies that the effect is not contingent on the duration of the spell -- It is not 1 "while this spell is active", like the way the headband sets your intelligence to 19 "while you wear this headband". It is just set to 1. Without interference, your intelligence will stay 1.
When the headband is considered, your intelligence becomes 19. You can wear the headband because that is not the same as activating it, which the spell prohibits. If the headband's wording used "when" or "after" you put the headband on, we could conclude that Feeblemind would win out, as it was the latest effect on your intelligence. But because it uses "while", we treat it as a continuous effect.
Can I communicate, understand language, cast spells, and activate magic items?
Because of the wording of the spell, you still cannot speak, cast spells, or anything like that. The spell does not give a condition on the secondary effects' end, other than the spells that can dispell it, so we must conclude that the effects are ongoing as part of the spell itself, not because of the mental scores of the target. It is not because of the intelligence score that you cannot do those things, they are separate from each other.
Furthermore, Intelligence is not described as being important for language. Page 12 of the PHB states that Intelligence is a measure of mental acuity, information recall, and analytical skill. On page 17 of the PHB, you know certain languages "by virtue of your race". You also acquire some from your background. In the DMG page 278 for the language of a created monster, there are no ability score requirements for language. There are some spells that don't work on creatures of poor Intelligence, some of them regarding communication, but other than that there is no other requirement to be able to understand or speak a language. If you want more reasoning, there is a very good answer for a similar question I posed regarding Intelligence and Language
We must treat the inability to understand languages, activate magic items, etc etc, as an effect that the spell is having on your character, not as an effect the low intelligence is having on him/her. Call it "brain fart on steroids" if you want. You're smart, but you just can't do certain things. Your Intelligence is 19, you're very good at recalling information and can make good saving throws, but you cannot cast spells, activate magic items, understand language, or communicate in any intelligible way.
Best Answer
The DM shouldn't do this
RAW seems to allow your interpretation.
However, this is not how it's intended to work
ASIs are class features, same as a Rogue's Sneak Attack or a Barbarian's Rage.
There is no precedent for spells or abilities that are able to permanently deprive a player of a class feature. It would have a huge impact and would definitely deserve its own clause in the spell text, were it the case.
It's a good way to alienate your players
ASIs are powerful features and are expected by the players, since class features are usually guaranteed upon leveling. Denying them an ASI just because they failed a save might be enough to put them off the game (especially if they are the min-maxing type).