This is a great question. I had to think about it for a while! AmE native, btw.
"As if you had your teacher with you" is definitely grammatically correct and sounds natural when spoken. If you're erring on the side of caution, certainly go with the "had" version.
If we venture off the grammatically correct path and wander a bit into the forest of sounds-okay-when-spoken-but-maybe-don't-say-it-to-a-pedantic-grammarian-or-put-it-in-a-term-paper, the "have" version might be acceptable as well. It somewhat accents the present-tense aspect of the phrase. It might connote the feeling "this course is so good it'll feel like your teacher is with you all the time, very much in the present".
Note that this feeling is by no means a hard and fast rule and different English speakers, especially the purists among us, might beg strongly to differ. However, in spoken American English, many good speakers might not bat an eye at the "have" version. Whether the English language company that wrote the ad put this much thought behind it or not is another question entirely.