Tobias
I have not used dump config to do database and transaction log dumps.
So you may have to do some digging to exploit what I explain below 
We control the transaction log dump file names in the sp_threshold procedure which can be activated by log thresholds or can also be called in a schedule by providing proper arguments.
When you install ASE SAP/Sybase provides a skeleton of this procedure which you can enhance.
bigdatetime allows you get time upto mircoseconds.
E.g. experiment with functions getdate(), and current_bigdatetime() to find out the difference.
Also datepart(us, current_bigdatetime()) allows you see the microseconds.
Using this you can generate a unique file name something like this
<path to archive directory>/dbname.yyyymmdd_HHMMSS_xxxxxx.tdmp
where xxxxxx is microsecond part.
Do not forget to pad microsecond part with zeros if you get less than 6 digits !!
And finally just be sure to get a unique file name add string converted from a random number to the name.
In dump transaction database command you can specify the full file name where the transaction dump will go.
HTH
Avinash