During Past Simple or Present Perfect?

After a verb that focuses on an activity, "during" is used to indicate a period or range of time for the activity. The meaning of the verb expresses a durative action or state. Using present perfect tense with "during" is awkward. We had a great time during our stay there.

Can I use present perfect with during?

Yes, you can. "During" is not an issue.

Should I use simple past or present perfect?

Use the simple past when the action started in the past, finished in the past, and is not continuing now. Use the present perfect when the action started in the past and is continuing now. The simple past tells us that an action happened at a certain time in the past, and is not continuing anymore.