turn on lights when alarm armed?

If a 5400 is running hot then you definitely need a relay, a smaller 10 amps capacity should be fine such as DEI 8166, I think that's the part number, I prefer to buy them raw, from the UK equivalent of Farnell and diode and wire them up myself, it costs a third as much!

Try Newark, part number 74R1537, IMO SRM-1C-SL-12VDC, PC relay, 10 amps power handling as a SPCO or 12 amps switching, $1 plus a few cents for a diode and some 30mm (1.25") shrink sleeving.

The current draw on the reverse lights is 3.88 amps.  That diode gets into the 170F (77c) degree temp range for the 30 seconds the 528 is activated. 

Should I be going to a 5 amp diode and can I use a 30/40 relay?

When all else fails, Read the Instructions

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Yes of course although that's an expensive route, that's why I suggested mini relays.

Issue resolved with a 6 amp diode. Work great!  Diode only gets warm for a 40 second timed activation.When all else fails, Read the Instructions

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I found this quite interesting for the following reason. In Europe (actually everywhere bar North America) we use amber direction indicators front and back. When I do a UK Cat l alarm and add a remote start, it doesn't feed these lights. So I modify using the GWR status output via a relay and a pair of 1N54XX diodes to each white light wire, (those white wires are on separate circuits here). Now these have a load of 3amps constant, ie 36watts. Each side adds up to 2 x 21 watts (front and rear) and 10watts (front wing side indicator) = 52 watts but I've never had any problems.