Fetch and read SQL Error Logs on AWS RDS (or any SQL Server, but it's meant for RDS where permissions are limited!). See natethedba.wordpress.com/quickie-read-sql-error-log-in-aws-rds .

FetchAndReadSqlErrorLogs.sql

USE master;
DECLARE @CONST_P2 int = 1; --Error log, not Agent log.
DECLARE @NumLogFiles int = 30; --SET THIS APPROPRIATELY!
--^ (there's no way to get it programatically outside of registry-reads/other-xp's, none of which is doable in RDS, AFAIK.

--Example: show events from 7/19/2018 between 2am and 2:30am PDT
DECLARE @StartDate datetime = '2018-07-19T02:00:00'
        , @EndDate datetime = '2018-07-19T02:30:00';

CREATE TABLE #ErrorLogs ([LogDate] datetime, [ProcessInfo] nvarchar(50), [Text] nvarchar(max));
DECLARE @LogNum int = 0;

WHILE (@LogNum <= @NumLogFiles)
BEGIN
    INSERT #ErrorLogs (LogDate, ProcessInfo, [Text])
    EXEC sys.sp_readerrorlog @p1 = @LogNum, @p2 = @CONST_P2

    SET @LogNum += 1;
END;

ALTER TABLE #ErrorLogs
ADD LocalDate datetime;

UPDATE #ErrorLogs
SET LocalDate = CONVERT(datetime, LogDate AT TIME ZONE 'UTC'
                                          AT TIME ZONE 'Pacific Standard Time');

SELECT el.LocalDate, el.ProcessInfo, el.[Text]
FROM #ErrorLogs el
WHERE el.LocalDate >= @StartDate
  AND el.LocalDate <  @EndDate
ORDER BY LogDate