Can I modify a slightly too big bike? A couple of months ago I went for a professional bike fit on a jig. They were struggling to find a bike to fit me perfectly (5ft1 female). They ended up ordering me a cyclocross bike.

The bike has cut time off my commute and is a dream...until I have to stop or start. I have absolutely no clearance between the top tube and my crotch. My feet touch the floor with difficulty when standing over the bar. I'm wobbling all over the place when stopping and it feels quite dangerous in traffic, I thought it would get better with practice.

Can I get smaller tyres, would this make my bike slightly lower? Or is this something I need to get used to and practice? Are there any modifications I can do? I'm a complete beginner so no idea if this is normal, I rode a rickety 25yr old mountain bike before this.
Try stopping with one foot down, leaning the bike over and leaving your dominant leg on the pedal with the pedal forward and up ready for a power stroke. By leaning the bike over you can get lots of clearance. I can often even remain on the saddle.

When you're ready to go, push off with your non-dominant leg (which is touching the ground) and give a strong pedal stroke with your dominant leg.  This will give you and initial burst of speed which will make balancing easier. Take a moment to adjust your position then continue pedaling.

I think frame stand-over clearance is a bit of a myth.  An older frame I use (older road bikes had very little clearance) has almost no clearance under my crotch and I really don't notice as I never find myself standing over the top tube.