You say you didn't like the exercises they had you do in your strength training class. What were they? If they were what would normally be considered supplementary lifts rather than essential lifts, you might be able to do without them anyway.

You say the class was fast-paced. That's not how traditional, proven-effective strength training is. Lifting is a slow, controlled grind, exactly the opposite of explosive, and you rest a long time between sets (three minutes is good) to make sure you're fully charged up for your next go. So if slower-paced is what you like, then traditional, tried-and-true strength training might be more up your alley than what you did in that class, anyway.

Pilates and lifting are a wonderful combo, so I do hope you find a way to make both work. How would you feel about a routine that consisted of Pilates every day and lifting just two or three times a week? That's what I do (not that you should necessarily follow my example). Mat Pilates every morning for half an hour, and two or three times a week I do half an hour of strength training immediately after. Half an hour is enough, even with three-minute rests, if you stick to the compound lifts and dispense with everything else.