To add to the confusion, the 7300 is also reported to have an "RF Tail." Oversimplifying, it continues to produce output even after the "key" line is opened at the end of a transmission for a very short time. This MAY adversely affect the relays in an amp, especially when using CW, since they may be being hot switched. The Groups.io site for the 7300 has a very long thread on this possible problem.
Tail should not be an issue if amplifier has proper sequence of transition from transmit to bypass/receive. First bias should be removed from power transistors, closing them down, then switch antenna to bypass line, finally switch input to bypass line. Or delay switching altogether until input RF will drop below threshold. That way there will be no hot switching anywhere. It is hard to understand by people stuck in 1950s vintage simple tube circuits. With timings involved designer does not even need to use FPGA - modern microcontrollers have enough speed to handle all that in software.
Dunno about SS amps, but the correct sequence when going from TX...back to RX on a TUBE amp is..(A) activate cut off bias. (B) RLS input relay. (C) RLS output relay. When going from RX to TX, the sequence is reversed, with C-B-A ( output /input / bias relays operated).
In actual operation the above preferred sequencing is a moot point. The 3 x relays can be operated in any sequence. I use the TX inhibit in the yaesu...and GRND the TX inhibit lead..... dead last. Zero RF coming out of the xcvr.... until AFTER the 3 x relays in the amp have already engaged fully..... including any contact bounce.
To pull this off, a short delay (2-20 msecs, depending on TR relay setup in amp) is used to GRND the TX inhibit lead, when going from RX to TX. When going from TX...back to RX, you want as close to a zero msec delay as possible. That kills the RF from the xcvr asap. You also want close to a zero delay on the TX inhibit lead, if using vac relays / mech relays, so when using VOX SSB, and TR relays in amp have 1/2 dropped off... (in mid air)..and VOX re-activated, the TX inhibit lead needs to be opened off, asap. With vac / mech relays in amp in mid air, their operate time is much less. Bottom line is... the last thing you want is bias activated, drive applied..and no ant connected. A kw into a wide open is bad news. On a tube amp, you will arc the load cap.
To solve the 'tail' issue, is an easy fix. On Alpha amplifiers, if the key line is pulled, while you are running say a 1.5 kw cxr, the relay's / Pin diodes in the Alpha will not release. They won't release if RF is still detected on the input of the amplifier.
If you look at W7RY's QSK mod for various tube amplifiers, he too, offers that 'RF detect' feature. It amounts to just 4 x components on his input TR relay board.... worth a grand total of 75 cents. If RF present, including a tail, the RF detector applies the rectified vdc to the main QSK board, and keeps the relays in the amp operated. (Input, bias, output). It's a dead simple solution, that's easily implemented.
Some older Icom xcvrs had real BAD overshoot, like the 706, and 775, and several other's. The 'fix' for those xcvr's was just a simple 9 vdc battery across a 500k to 1 megohm pot. - 1 to -9 vdc comes off the wiper..and applied to the ALC input on the xcvr. The alc voltage has already been derived externally, and is always sitting there. No overshoot. Dial up 86 watts, and all you get is 86 watts.
That doesn't help the OP, with his older K3..and no ALC input on the K3. The best compromise would be to use a small 2-3db pad on the output of the xcvr. Or increase the input padding inside the IC2KL by 2-3db.