If I may be so bold I'm going to break up what you are asking into different chunks and rephrase it.
1)What do I do with my rental income (the >4% you mentioned)?
2)Should I pay down my mortgage?
3)Should I lump sum invest or dollar cost average it over time?
I'll answer in that order, and please let me know if I misunderstood anything.
1) The rental income could go into whatever you want, towards living expenses, towards investments, anything. It's extra income. It all depends on your long term goals.
2)This depends on your risk tolerance, the interest rate on the mortgage, the potential returns on investments, and your long term goals. There is no clear cut answer. The best rule of thumb is that if you can get more from a potential return on investment do that rather than pay down the mortgage.
3) Lump sum investment will often perform better, DCA is advantageous in the scenario spoken of in #1 when you excess cash monthly to invest you are practicing DCA. I would personally put it all in now but it depends on your long term goals (common theme I know).