It is possible to get DTM to re-evaluate page load and event based rules after page load, but it involves some internal _satellite method calls passing some dummy/rebuilt objects, it is not supported by Adobe for public use, and in practice it won't actually work for you 9/10 times.
Given the current limitations / state of DTM, the best way of doing it is along the lines of what Jantzen.Belliston already mentioned.
But in general, one of the benefits of migrating to Launch is that it is a lot more friendly to situations such as yours.
Okay well to be clear.. depending on what the full details/requirements for the tags vs. the site are, you may not have to do a full page refresh to make the tracking trigger. There are alternatives to "refresh the page" and "artificially re-invoke/eval PLRs"!.
For example, Let's say you have a page load rule (PLR) with a condition to output if path is /somepage.html AND if a selected dropdown option value is "bar". And let's say it is possible for the page to be loaded with "bar" pre-selected, but it's also possible for the dropdown to start with a different option selected. But you only want the tag to be output if "bar" is selected, be it initial page load or after page load.
You can create your PLR as you have already done. Then, you can create a separate event based rule (EBR) similar to your page load rule conditions, where you set a condition to ensure you are still on the same page /somepage.html, and have the event type be forms > change. The biggest downside to this is having to maintain two separate rules for one tag.
Alternatively, you can leverage an EBR of dataelementchanged type to have a single rule to handle both page load and after page load scenarios, but it's a bit clunkier and potentially not as accurate (depending on site function vs. poll rates). I wouldn't really consider it unless I'm doing a lot of tagging/rules and maintaining duplicate rules for a ton of tags starts to outweigh the the dataelementchanged approach.
As for an event rule, they are typically used in the exact opposite way. An event rule watches for something to happen on the page (including your own custom event), then fires. If you want to trigger an event based rule based on some custom code, I'd suggest firing your own event or using direct call rules. Here is a article that talks about using custom events in DTM