Pretty simple huh?
The problem is that people aren't "strict" enough in their lifting. They let their backs bend, they lean back, they do all sorts of things.
A nice strict press will have a nicely maintained back position. Ideally, the necessary leaning back should occur at the ankles as should the return to a vertical body under the bar.
If you practice anything else, you are merely practicing how to compromise your spinal stability when the load is too high for you.
So start with the bar and slowly load up. I will guarantee that you will begin to lose form well before your current 1RM. You should consider your new 1RM "strict" strict press as the best way to practice your handstand push ups...and don't forget, it is the slow eccentric phase of the strict press that will help develop your strength so don't just dump the bar, keep it under tension all the way down onto your shoulders!
Training this way will improve your freestanding handstands because you will be able to control your body and keep all the joints stacked on top of each other during the movement - that will improve efficiency and stability and ultimately your total output!
Here is a pretty video of Dave Durante doing free standing strict handstand push ups to Rich Froning's Bring Sally Up challenge...amazing strength, stability, flexibility and control!