I need to get fertilizer out and spray my broadleaf weed killer this week (pre-emergent already down).

Any issues doing both today or should I space them out by a few days?

You can do them both. Some will tell you it's better to space them apart, and it may be, but you're not hurting anything doing them together.

Nothing really wrong with doing both but you will see better results if you fertilize first, wait a week and a half and spray or broadcast post emergent. This boosts the metabolism of your weeds which helps the uptake of the weedkiller. Provides you with a more effective kill and to my eyes it clears out weeds faster.

I fertilized at the beginning of March, then a week later hit the yard with a weed killer. Then just this past weekend, had the lawn core aerated and hit it with another fertilizer right before some light rains came.

Expecting some good results...