Poor drainage sounds like the problem, so how about a shallow system of ditches, two or three of them, 30cm deep, 20cm wide, diagonally across the lawn, with and across the slope (and there's bound to be a gentle incline), back-filled to half the depth with 20mm shingle, then a membrane laid in to stop the shingle clogging and filled up with topsoil and re-turfed. That should help the water drain away, and improve the soil condition, which would then improve the health and vigour of the grass. Phil J

Phils response above is the full on belt and braces approach and if you can face it do it :)

A method which may not work as well as Phil's but should improve things is as follows (From a response I wrote to a similar question on another forum:-)

Moss is caused primarily by bad drainage and to a lesser extent shade. Scarifying and applying moss killer will temporarily remove moss but it will reappear if the underlying cause is not dealt with.

To improve poor drainage aeriate the lawn. You can use a garden fork stuck into the lawn at 4 inch intervals and waggled about to increase the size of the holes. Apply lawn sand which will fall down the holes and leave semi permanent drainage channels.

It is possible to hire a machine to do this as ,with anything other than a very small lawn ,it's hard and tedious work!

Scarification of the lawn using either a leaf rake or a mechanical scarifier will remove dead grass and moss .This gives the grass chance to grow well. Don't set the mower too low,particularly in early spring and autumn.

Also feed the lawn with a spring weed and feed in spring and an autumn feed in autumn.( They have different chemical compositions. The spring feed will make the grass grow and the autumn feed is designed to improve the roots.)

You can now buy grass seed grown to cope well with shade

You have said your landlord is not happy for you to returf, so I doubt he'd be happy if you really tried to address the problem in any realistic way, such as suggested by Phil above. Not much you can do with a difficult landlord, really, other than NOT treat the moss, let it grow - at least its green most of the time and soft to walk on.
You're right about not cutting seed with a lawnmower - if you do decide to seed, treat for the moss, rake it out, wait 6 weeks after treating, then rake up the bare areas to a fine tilth (friable, loose inch or two on the top) spread the seed, and then cut those areas with shears for the first 2 cuts. Unless you have a hover mower, particularly a flymo - these will cut seeded grass without damaging it as long as the blade is sharp. Treat for the moss in late winter/early spring, and if the ground is dry enough 6 weeks later, prepare and sow the seed.

ive even used good scissers to cut the grass the first time especialy if it isnt a very big lawn .

Funnily enough I've never had any problem cutting grass seed with a lawn mower. Two of my customers laid new lawns this year and I used a very heavy duty self propelled mower with a roller with no ill effects. I didn't cut the grass until it was about 3 inches tall and then took off only an inch each week( leaving two inches) The lawn has established very well.

Obviously though if you can use a hover or light weight mower then use that.