The SRD is very much on your side as a Fighter/Bard if you're wearing light armour:
Bards can wear light armor without incurring any arcane spell failure chance for their bard spells.
And with heavier armour it's got your back too – it's explicit that you only suffer an Arcane Spell Failure chance with spells that have somatic components:
If the spell lacks a somatic component, however, it can be cast with no chance of arcane spell failure.
However, if this DM has been playing since before 3rd edition, they may have carried forward the "physics" of magic from earlier editions. As the DM this is their prerogative, but if that's the case then they didn't communicate this to you when it was important; if they had, you might have made different equipment choices, or even not chosen to multiclass in the first place.
However, telling the DM they're wrong will likely get you nowhere quickly – if the DM is using a "but my world works this way" house rule, discuss the mismatch of expectations and find out what would be an acceptable solution that would leave you with a character you still want to play while keeping the DM's world consistent. (Consider that the world is their character, and you do want your DM to still want to play it!) If you approach this conversation with self-assurance (but without self-righteousness!) and without putting your DM on the defensive, you'll have a much better chance at a constructive conversation with an outcome that you both find agreeable.
Yes.
Here are the rules for Arcane Spell Failure:
Casting an Arcane Spell in Armor: A character who casts an arcane spell while wearing armor must usually make an arcane spell failure check. The number in the Arcane Spell Failure Chance column on Table: Armor and Shields is the percentage chance that the spell fails and is ruined. If the spell lacks a somatic component, however, it can be cast with no chance of arcane spell failure.
The rules do state a few exceptions (spells without somatic components in the quoted portion, and bards getting light armor without ASF elsewhere in the linked document), but proficiency is not one of them.
In other words: Even if you are proficient with armor, you still suffer its Arcane Spell Failure chance.
Even though it says that a magus can cast magus spells while wearing light armor without incurring the normal arcane spell failure chance?
Yes.
Kensai is an archetype of Magus. The Magus does have a class feature that allows them to use light armor without spell failure:
Weapon and Armor Proficiency
A magus is proficient with all simple and martial weapons. A magus is also proficient with light armor. He can cast magus spells while wearing light armor without incurring the normal arcane spell failure chance. Like any other arcane spellcaster, a magus wearing medium armor, heavy armor, or a shield incurs a chance of arcane spell failure if the spell in question has a somatic component. A multiclass magus still incurs the normal arcane spell failure chance for arcane spells received from other classes.
But this is entirely replaced by the Kensai class feature of the same name:
Weapon and Armor Proficiency
A kensai is proficient in simple weapons and in a single martial or exotic melee weapon of his choice. A kensai is not proficient with armor or shields and suffers normal arcane spell failure chance when casting magus spells while armored.
In other words, the class behaves as though the entire "Weapon and Armor Proficiency" block had been removed from the Magus, and replaced with the one from Kensai.
The Kensai will also not get the Medium Armor (Ex) or Heavy Armor (Ex) class features of the Magus class, as these are replaced by the Iajutsu (Ex), and Iajutsu Focus (Ex) class features.
Best Answer
It seems like the Eldritch Scion Magus archetype is what you're looking for. It's basically a Magus that can cast its spells spontaneously. It also loses out on some other class features in order to gain access to a bloodrager bloodline. Like a regular Magus, an Eldritch Scion can use light armor without spell failure at first level, medium at 7th, and heavy at 13th.
You might also look at some of the third-party magus archetypes that grant spontaneous spellcasting: Extempresario, Spiderhawk, Primagus.
Summoners also have the ability to cast spontaneous arcane spells in light armor right away, no archetypes needed. You might also consider taking the Synthesist archetype, which replaces your eidolon with a suit of magical power armor.