Here are the results of my investigation into why the Edmonton IRLP node can not be accessed using a Radio Shack HTX-404 handheld UHF radio because of DTMF tone problems.

The Edmonton UHF IRLP node requires the use of a 103.5 Hz CTCSS tone for repeater access.
When a DTMF tone from a HTX-404 is received by the repeater, the repeater will drop out momentarily after approximately 200ms of tone transmission before repeating the remainder of the tone transmission.
The repeater also drops out momentarily after the end of each DTMF tone transmission. The drop outs are due to the repeater CTCSS decoder momentarily losing CTCSS tone input at the start and end of each HTX-404 DTMF tone transmission.

The CTCSS tone decoder output is used by the IRLP interface for the receiver COS signal in order to keep repeater hang time, courtesy tones, and I.D.'s from being transmitted over the IRLP system. As a result, what the IRLP interface receives, when a HTX-404 is used to enter the 4 digit IRLP access code, is 8 individual single tone transmissions.
This is a recording representative of
DTMF tones received from several different handheld and mobile radios that I own.

All recordings are of a
CTCSS 103.5 Hz tone of 600 Hz deviation and a DTMF tone of 3.5 kHz deviation.
Notice the clean transition from
CTCSS only to CTCSS + DTMF tone.
The return transition from
DTMF + CTCSS tone to CTCSS only
is also smooth.
This recording is of a DTMF tone received from a Radio Shack HTX-404 radio.

Notice the extreme modulation swings at the start and end of the DTMF tone.
The transition from
CTCSS only to CTCSS + DTMF tone first results in a deviation of more than -15 kHz and then +12 kHz before returning within the normal 5 kHz bandwidth of a receiver.
The return transition from
DTMF + CTCSS tone to CTCSS tone only first results in a deviation of about +12 kHz and then -5 kHz before returning to normal.

It is now obvious that the CTCSS decoders are dropping out due to the loss of received CTCSS tone during the period that the HTX-404 is over deviated.  A similar CTCSS decoder in the IRLP link radio reacts the same way as the repeater CTCSS decoder does. Several amateur transceivers tolerated the HTX-404 signal much better, but also occasionally drop out. The repeater and link radio CTCSS decoders could be possibly modified to accept longer CTCSS signal drop outs, but that would lead to unacceptably long system turn around times.

The problem with the HTX-404 is in its design. The DTMF encoder I.C. signal output pin is normally biased at 0 volts. While a DTMF tone is being generated this pin is biased at approximately 2.5 volts. The sudden shift to and from 2.5 volts, which is transferred through the audio circuitry via coupling capacitors before being applied to the varactor VCO diodes, creates the wide VCO frequency swings. This is a poor design that a few cents of components could have corrected. I tried to come up with a practical solution for the problem, but with SMT construction and limited space in the handheld, any solution would have been messy and require a lot of time and patience to install.

I also found that the tiny push-button PTT switch, on the HTX-404 I tested, generated noise on the transmitted signal if not held very still. This switch should not have been creating any problems as the radio was nearly new.  Again, a few additional components in the design could have eliminated this problem. This switch noise problem also affects the radio in exactly the same way as the DTMF encoder does. And since the noise pulses occur much more rapidly then DTMF digits would ever be sent, much more of the transmitted CTCSS signal is lost and even the more tolerant radios that I tested, when utilizing CTCSS squelch, had a hard time receiving any audio from the HTX-404.

CONCLUSION:

DO NOT PURCHASE A RADIO SHACK HTX-404
IF YOU NEED OR EVER PLAN TO USE CTCSS ENCODING.

Otherwise, I liked the radio. It is solidly built and I would even buy one if:
I needed a UHF handheld,
the price was right,
and
I had NO NEED for CTCSS.