I'm trying to prepare new router before switching ISP router (which I have no access to) to bridge mode.

I know its totally not recommended but what would happen if I configured DHCP pool on second router (which is behind the first router) to be still on same network but out of range of first DHCP pool? For example first router's DHCP is 192.168.1.2 - 192.168.1.99 then I would set DHCP on second router to: 192.168.1.100 - 192.168.1.254 ? The first router would have only one DHCP client which is... the second router and whole network would be actually under the second router for a while?

Pools shouldn't go through subnet boundaries.
You can do something like the following.

Set a static IP address on the Wan side of the second router.

One of the following should be fine.
192.168.1.2/25 (0 to 127) gateway 192.168.1.1
If only have 2 devices on Wan side of second router.
192.168.1.2/30 (0 to 4) gateway 192.168.1.1

Note: The first router still thinks it is connected to a /24 subnet,
but it only has 1 device to connect too.

masquerade enabled on WAN port of second router.

On the Lan side of the second router, set IP address of 192.168.1.129/24
Set the dhcp pool to be from 192.168.1.130-192.168.1.254
Could also allow room at both ends for servers.

Next:
Enable proxy arp on second routers lan bridge interface.

Efficient mode:
Setup the dhcp server on the lan side with gateway address of 192.168.1.129

Less efficient, but more long term compatible:
Set the dhcp server on the lan side with gateway address of 192.168.1.1
Devices on lan side will arp for 192.168.1.1, second router will reply with its mac address because of proxy arp...

Later:
When remove first router,
Remove proxy arp from router lan Interface.
Change ip address of router to 192.168.1.1
(if required)
Change dhcp server gateway address to 192.168.1.1
Change dhcp pool range.

Next:
Enable proxy arp on second routers lan bridge interface.

I don't quite understand proxy arp part, I see it mostly used in VPN setups, what would it do here?

Efficient mode:
Setup the dhcp server on the lan side with gateway address of 192.168.1.129

Less efficient, but more long term compatible:
Set the dhcp server on the lan side with gateway address of 192.168.1.1
Devices on lan side will arp for 192.168.1.1, second router will reply with its mac address because of proxy arp...

Long term compatible in a way that I wouldn't need to change it back when I remove first router?

Less interested in hypothetical questions more interested in understanding the requirements.
You have an ISP router which you cannot control but you stated going to put in bridge mode.
Q1. Can we assume you will get then a public WANIP from the bridged router?

Now you talk of two routers. Is this two other routers besides the ISP one?
The mikrotik MAIN you are installing after the ISP bridge mode device and another mikrotik device behind the MAIN router?

The reason I ask is that you state you are putting the ISP router in bridge mode or at least getting the Company to put ISP in bridge mode, which then one would assume you are getting a public IP and thus any talk of being on same private DHCP as the first router would be nonsensical and thus you must mean having a third router having same DHCP as the second MAIN router.........

Right now I am assuming your MAIN mikrotik router would get a private IP from a subnet on the ISP router and this is useless since you cannot forward ports etc.........

Even the simplest network diagram in such a situation is worth more than 1000 words.

viewtopic.php?f=3&t=45259

Too late bartoz, nobody reads such things before posting. When will the little green and red men learn..................
You have the saying in poland.
Zamknij drzwi stodoły, gdy koń ucieknie

And another for you:
Polski turysta wraca do domu po wizycie w ZSRR. Niesie dwie bardzo duże i ciężkie walizki. Na nadgarstku ma nowy zegarek produkcji radzieckiej. Mówi celnikowi: „To jest nowy radziecki zegarek. To cud nieznany w krajach kapitalistycznych. Widzisz, pokazuje czas, tętno, fazy Księżyca, pogodę w Warszawie, Moskwie, i Nowy Jork, i coraz więcej!”

„Tak, to cud” – zgadza się celnik. – A co masz w tych wielkich walizkach?

„Och, to tylko baterie do tego zegarka”.

The mikrotik MAIN you are installing after the ISP bridge mode device and another mikrotik device behind the MAIN router?

There is only 2 routers: main ISP router and Mikrotik. I want to temporarily set Mikrotik as second router, see if its working, then make it main router with least amount of settings to change.

I understand, you have a bunch of equipment tied to the ISPs router and lets say the router gives out IPs of 192.168.10.X,.

YOu want to setup the MT router now so it gets its wanip as 192.168.10.Y but also still keep the LAN devices with their respective current IPs.
This causes issues and not recommended, what does it matter what IP addresses your current devices get??
Once you assign them in this variation with something like 192.168.100.X when you put them in NAT behind the MT router, then you will not have to change them
when you give the MT a public IP vice a public IP.

At one point or another a change will have to be done, the earlier the better.

(...) what does it matter what IP addresses your current devices get??

Because there are some devices with static IP in 192.168.1.0 network and I don't want to reconfigure them now.

Then simply wait until you get the ISP router bridged...........