3.4.21 — after a verb whose agent is the same [as the main verb's], in prior time. Two conditions, both in that one compound word: समानकर्तृक (same agent) and पूर्वकाल (prior time, not simultaneous). Violate either and क्त्वा is simply the wrong word — not stylistically weak, not unusual, wrong.
Same agent — the ordinary case
"Having eaten, [he] goes" — the Kāśikā's own example on this sūtra. One person, Devadatta, both eats and goes; the eating is finished before the going starts. Both conditions hold, so क्त्वा is exactly right.
Same agent, violated — the trap
Attempting "the Brāhmaṇa having eaten, Devadatta goes" — but as written this can only mean "Devadatta, having eaten, goes." The Brāhmaṇa's meal has no way to attach.
"The Brāhmaṇa having eaten (locative absolute), Devadatta goes." Two different agents need a locative absolute — taught below — not क्त्वा.