Has anyone been able to figure out how to use the Fire Tablets as a speaker / chime? In the past I was able to use Big Talker and a few other tools, but with the new Edge transition, it seems all of that functionality has been broken? Any leads or suggestions would be much appreciated.
Do the Fire Tablets support Alexa routines? Can the ST skill be installed and your devices sync’d from ST to Alexa? If so, you can use virtual devices in ST to trigger Alexa routines to perform announcements. For example, I have Alexa greet us when we arrive home on our Echo Show.
Which model fire tablet do you have? There is one fairly easy option, but it’s pretty limited, it does require one of the newer fire tablet models, and you also have to have a second smart speaker. So this may not meet your requirements, but I thought I would go ahead and detail it.
If you were asking about an echo dot or other echo device, as @h0ckeysk8er mention, that’s actually quite easy. There’s an option called “announcements“ under Alexa routines and that let you select from your echo devices to speak a custom phrase when something else happens.
Or you can choose from sound effects, including chimes.
Unfortunately, the fire tablets are not available to select from with that method. 
However, if it’s a newer fire tablet, it’s plugged in, and you have it in “show mode“ you can make it to speak a custom phrase using the “Alexa says” option— but only when you speak a custom phrase to it.
So that’s where the second echo device comes in.
You can set up an Alexa routine so that when an event occurs (including a virtual sensor opening), the second echo device will speak a custom phrase.
And then you have a second Alexa routine so that when the tablet hears that custom phrase, it speaks another custom phrase.
This works, I tested it this morning, but obviously it’s only useful for some specific situations and it does require that second device. And it requires that the tablet be plugged in.
Also, at that point, you have to wonder why you’re using the tablet at all, since you have to have the second device in the same room already. (because the tablet has to be able to hear the spoken instruction from the second device.)
So I don’t know if that helps any or not, but I thought I would mention it.
It certainly feels like it would be a lot easier to just wait until an echo Dot went on sale and then use that instead of the tablet. You’d get more feature options (including chimes), and a simpler setup. 
You might be able to use the Custom action to simulate having Alexa “speak” the phrase. I use that for both my Ecobee and Denon HEOS Home Theater Receiver. If so, a routine with the Custom action might be able to trigger the Alexa Says routine.
Tried that, it doesn’t work, because you can’t direct it to the tablet individually. And that’s the only way to get the tablet to respond with the spoken phrase.
Custom actions can activate an Alexa skill, which is how it works with your ecobee, but getting the tablet to speak is not using an Alexa skill.
I’m not sure why the tablets are so limited in what you can do with them unless it’s a concern about battery life, but for right now, they are. You can’t even include them in “all devices“ when making an announcement. 