Flora also make one that is alcohol-free
I notice you ask about a way to go without boiling water but to my knowledge there isnt a way to avoid that route if you have a tincture containing alcohol. You can put a dropperful in about an ounce of water that has been boiled and it will neutralize the alcohol. Alcohol will kill gut flora but in very small amounts it wont have much of an impact. If you are really sensitive and have the option it is better to use an alcohol free formulation. The larger issue with alcohol is that it will feed yeast like wildfire if you have an issue with that, and also is a gut irritant for people with gut issues.
Take half a dropper or spoonful of the bitters and hold it on the back of the tongue for a few seconds just prior to or after each meal to stimulate the vagus nerve and production of stomach acid for those with weak digestion or digestive issues. I have tried this but not consistently enough to produce any noticeable results. I have found fermented vegetables eaten with protein meals to work better for me to aid digestion.
I would suggest taking your probiotic on an empty stomach first thing in the am, with just water and one hour away from any other food, drink, or supplements. If you want to follow up with a second daily dose of probiotic you can take it under the same conditions, before bed - as long as it is at least 3 hours after you have finished your last meal (on a reasonably empty stomach).
I couldn't argue specifically whether or not bitters, probiotics, or any other supplements are paleo but am more concerned with whether or not they can help to improve digestion or achieve other wellness goals.