That 350 will absolutely tow you safely. The chassis, payload, and axle ratings are much more important than how much HP and TQ you have. That 7500 lb rating, IMHO, is a performance rating. I would worry more about the 4l80e transmission than the motor.
If it were me, I would ad a second transmission cooler in line with the factory external cooler since these model suburbans often suffered from a lack of transmission cooling capacity. Use a B&M or a Derale stacked plate model. They are very nicely built and perform great.
I would also recommend a transmission temp gauge so that you can monitor the temp in case it starts to overheat.
Other than that, enjoy your suburban!
Thanks!
Jeremiah