You will need to enable Send Logging in your MID. It is a free addition but if it is not enabled you will need to create a support ticket.
You then create a SendLog DataExtension using the SendLog template. Add any columns you may want to link to your TriggeredSend DataExtension(s) -- to link them the column name in your TriggeredSend and SendLog DataExtensions must be named identically. You could have multiple columns if you desire such containing SubscriberID or whatever else.
When you send an email using the TriggeredSend, a record will be created in the SendLog DataExtension automatically. It is linked to your TriggeredSend DataExtension record with that column mentioned above. It also contains things such as the TriggeredSendDefinitionObjectID, BatchID, SendID, and a couple others.
Once the above is completed, you can then begin using the SOAP Api to request BounceEvents. The BounceEvent does not have everything you need, so youll need to use the information from the BounceEvent (BatchID, SendID, etc.) to then use the SOAP Api to query that SendLog DataExtension in order to determine which send for which recipient bounced.
It is not the most elegant solution but multiple Retrieve requests for different objects using the SOAP Apis is the only way to do it via Api as there are no equivalent REST Api endpoints.