Your GM is correct. Until you are higher level you are not able to cast higher level spells. The entire point of prestige class entry requirements is to gate entry to more experienced characters, and your interpretation basically reduces to just saying "an arcane spellcaster" which if that's what they meant, it's what they'd say. You need to actually be able to cast the spell, so you need appropriate level + high enough stat + correct class + not having swapped out your spellcasting ability for some other ability and so on.

Remember that this is supposed to represent some kind of in game qualification.

"Welcome to the Guild of Master Evokers! So you want to join, eh?"

"Yes!"

"OK, show us your stuff, cast something big - you know, a fireball or whatever."

"Well, I can't now - but I have the ability to one day!"

"What does this look like, Hogwart's? Don't let the door hit you in the ass on the way out."

In this context, "able to cast" and "ability to cast" mean the same thing. The designers' neglecting to use the exact same verbiage for every writeup is... pretty typical, actually, and not indicative of any deliberate nuance.

Also there seems to be a misunderstanding about a high stat possibly letting you cast higher level spells early - this is not the case. "In addition to having a high ability score, a spellcaster must be of a high enough class level to be able to cast spells of a given spell level." -d20PFSRD

This winds up getting away from the idea of the hulking thug rogue/barbarian; the solitary barbarian level is used to grant mobility and agility rather than strength and endurance. That said, this is a very vicious combat build with lots of dirty tricks, very capable of pouncing on opponents unawares and tearing into them in an unholy frenzy.

The Assassin

For a low-optimization, low-magic campaign, where you want to focus on stealth and skills but still be able to deal the hurt, you could do a lot worse than the core assassin prestige class. It gets a number of low-level spells, so it should not be a problem for low-magic, but if there’s little magic available, you definitely want what little you can get.

If alignment is an issue, consider the avenger: it replaces the Evil requirement with a Non-Chaotic requirement, which is often much more amenable to characters.

Spells

Spells are the reason you go this route. The assassin spell list is quite solid, for as small a list as it is, and it will likely fly well in a low-magic game.

This recommendation does assume that you have access, at the least, to Spell Compendium. The core assassin spell list leaves a lot to be desired. If you want to use the avenger, also make sure your DM is OK with adding the Spell Compendium assassin spells to the avenger list. That’s in line with the guidance in that book for non-core spellcasting classes on pg. 3, so it shouldn’t be a problem, but it’s a conversation you should have.

Psionics

The psionic assassin from Secrets of Sarlona also exists, and adds the very interesting option of using Expanded Knowledge for psionic minor creation, which is an excellent way to get poisons. That said, unless your DM allows other assassin spells to be converted into psionic counterparts, it’s not a great option. If you can, then you might as well?

Death Attack

Don’t focus too much on Death Attack; at best you can attempt it during a surprise round after spending three rounds studying your target out of combat. Attempting to hide and wait three rounds mid-combat is always a bad play.

Poison Use

Poison Use is meh; if you want to use poison, you want the Master of Poisons feat from Drow of the Underdark anyway. That said, Master of Poisons is a quite-solid feat, and Craft (poisonmaking) can get you poison at a huge discount (if you have a source of materials, ⅙ market price). Poisons provide a way to apply debilitating effects along with straight damage. See the Arsenic and Old Lace handbook for more details if you’re interested in going for poisons.

Again, if you are avoiding being Evil, you have to clarify something with the DM: poisons, according to Player’s Handbook and Dungeon Master’s Guide, are not evil, though usually illegal (and always dishonorable, which is why they are barred to paladins). Book of Exalted Deeds, however, says they are actually Evil – and then gives a definition/explanation of how and why they’re Evil that would include, for example, the natural poison of the couatl, that is, a notably Good creature. Book of Exalted Deeds is not a good book. Just make sure, if you don’t want to be Evil, that your DM agrees that Book of Exalted Deeds is stupid, and poisons aren’t any more inherently evil than swords.

Hide in Plain Sight

It’s a long time coming, but getting Hide in Plain Sight, along with taking the Darkstalker feat from Lords of Madness, will prevent a lot of the major ways to simply shut down your stealth. Highly recommended.

The Black Dog

This prestige class from Dragonmarked stacks with assassin levels for Death Attack DCs, and has a couple of excellent abilities aimed at poison. I’ve decided that poison is a good direction to take this, hence its appearance here.

The black dog requires the Mark of Hospitality feat, which in Eberron is limited to halflings. This is a matter of setting fluff, so if you aren’t playing in Eberron, the Mark of Hospitality might be available to half-orcs. On the other hand, I actually like halfling a great deal more than half-orc here. It does mean a shift from Strength to Dexterity which you may find undesirable, but half-orc is quite weak, and the halfling’s size is basically all to your benefit (average of −1 on damage rolls, but +1 attack, +4 to stealth, and +1 to AC). The existence of the strongheart halfling (Forgotten Realms Campaign Setting) really seals that deal.

Suggested Build

Strongheart Halfling Rogue 3/Barbarian 1/something 1/Black Dog 5/Assassin 10

At Rogue 2, the Penetrating Strike alternate class feature (Dungeonscape) is key, and at Barbarian 1, you really want the Lion Spiritual Totem alternate class feature (Complete Champion) to get Pounce instead of Fast Movement (you’re not a charger per se, but you do want to maintain mobility and full-attacks, and pounce is a really good way to do that).

I’d replace Rage with Ferocity, though I’d also consider Whirling Frenzy.

The something 1 I’m not sure about; barbarian 2 is OK but not great, rogue 4 is fairly mediocre...

Some options that should be no problem:

• Fighter. You want Weapon Finesse at this point, so that’s a use for your bonus feat, and the Hit-and-Run Tactics alternate class feature from Drow of the Underdark trades some armor and shield proficiency you probably won’t use for +2 to initiative and Dex-to-damage vs. flat-footed foes, which is awesome.

• Ranger. Simple: full BAB and 6+Int skills. Boring, but functional.

• Anything that gives Sneak Attack +1d6 at first level. Again, boring, but functional.

Some options that probably aren’t allowed, but could be awesome:

• Cleric. Best single-level dip in the game, bar none. Could be worth it even if you get none of its spells. Domain granted powers are awesome, and failing that you can trade them for Devotion feats (Complete Champion), many of which are excellent. Turn/Rebuke Undead is excellent for Divine feats, including the amazing Lolth’s Caress (Drow of the Underdark). While a halfling who worships Lolth is weird, with this build’s interest in poison it’s not a stretch to worship the goddess of spiders. Or if Lolth doesn’t exist in your setting, maybe you can just ignore that requirement.

• Ardent or Psion. One level is sufficient to manifest psionic minor creation, which will get you a plant-based poison that lasts an hour. Not a bad deal at all.

• Swordsage. Cloak of deception provides instant flat-footing for an enemy, once per encounter, while shadow jaunt provides great mobility. Wolf fang strike and sudden leap allow you to continue to dual-wield effectively even when you’re forced to move and cannot charge. If you later take Martial Stance, you can snag assassin’s stance for +2d6 Sneak Attack damage.

Anyway, you definitely want the Craven (Champions of Ruin), Darkstalker (Lords of Madness), Master of Poisons (Drow of the Underdark), and Weapon Finesse feats. Plus, black dog requires Mark of Hospitality (Eberron Campaign Setting).

The feats Extra Rage (Complete Warrior) and Wild Cohort could be very nice for you, if you can squeeze them in (maybe later, or if you take a Fighter level and have another feat to play with). Note that I don’t recommend Wild Cohort so you have a pet fighting alongside you: I recommend it so you have a reliable, cheap poison dispenser following you around. Unfortunately, both feats are kind of lackluster by the time they fit into the build I am proposing.

Anyway, your feats may look like this:

• Bonus racial feat: Weapon Finesse
• 1st-level feat: Master of Poisons
• 3rd-level feat: Mark of Hospitality
• 6th-level feat: Darkstalker
• 9th-level feat: Craven
• 12th-level feat: Extra Rage?
• 15th-level feat: Wild Cohort?

With this combination, your attacks hit hard, and you are hard to find. You have poisons that you can use selectively, and they are very dangerous. You have a smattering of spells that will make a huge difference in a low-magic world. Seems like a pretty solid approach to me.