Line frequency leakage current in SMPSUs isn't inherently due to it being a SMPSU, its due to the class Y capacitor from secondary 0V to the negative side of the primary DC bus, that is required as a return path for the HF leakage current capacitively coupled (by the inter-winding capacitance) from the 'hot' end of the primary to the secondary if there isn't an inter-winding screen.
If your proposed linear PSU's transformer doesn't have either a grounded inter-winding screen, or a split bobbin construction, it may still have enough interwinding capacitance to pass enough leakage current to be troublesome, causing its secondary to float at a significant AC voltage.
Unfortunately that transformer on EBAY doesn't look like it has a split bobbin construction and probably doesn't have an inter-winding screen. Unless it has comprehensive manufacturer's data, you'd have to buy it and test it to find its leakage current. If you are floating the secondary, and it doesn't have a proper datasheet, I'd also want to do a 1KV Hi-Pot test between the primary and secondary sides before I'd trust its insulation.
If the leakage current is excessive, the only option is to ground the secondary side 0V rail, which will result in a ground referenced output, though if you need it floating for DC, its possible to use a capacitor between 0V and ground(+ a series resistor to limit the surge if its ever hot-plugged).
If you haven't stated buying parts yet you may want to re-think the PSU design. The +5V rail doesn't need to be extremely quiet so a switching regulator can be used there, with heavy filtering on its output to minimise switching frequency breakthrough,. That means you don't need the 9V windings. I'd also bet you don't need a lot of -12V current, so you *may* be able to capacitively couple a bridge rectifier for the negative rail, so it can all run off a single 15V RMS secondary, though a 15V-0-15V transformer would be preferable.
Have you got figures for the load current required on each rail? If not, I would suggest measuring them using the existing PSU, with both channels analog outputs loaded with 50R 1W resistors (47R would be close enough), while outputting a maximum amplitude squarewave. Don't keep the load on at max output for longer than half a minute (the FY6600 short circuit overload rating).
For the primary feed I'd run the mains through a 1A slow blow fuse in the Line connection, then straight to the primary. If your mains supply is particularly dirty, you may want a suitably rated varistor across the primary. The transformer *should* have a non-resettable thermal fuse buried in the winding. If you add a mains power switch, don't use one with any exposed metalwork.
On the secondary side, use good quality rectifiers rated for at least three times the required output current, and fuse the unregulated rails after the bulk decoupling caps*, with fast-blow fuses rated for the expected maximum output current. To prevent damage to the load if a fuse blows and a rail is lost, connect three diodes between the regulated output rails in order of their voltage,each cathode to the most positive rail, so that if a secondary side fuse blows, the -12V rail cant rise significantly above 0V and the +12V rail cant drop below the 5V rail etc.
* Its possible to fuse between the secondaries and the rectifiers but you'd need four fuses, its *MUCH* more difficult to calculate suitable fuse values, and you'll either have to use slow-blow fuses or go to a significantly higher current rating to handle the initial surge when mains is connected and the bulk decoupling capacitors initially charge.