Your perceived enemy has a population of 300,000
Your total population is a miserable 5000 and your allies have only 15,000. The total population your side has is just 20,000.
You cannot out pace their growth.
More to the point this enemy could consume your entire country with just 60,000 people using a classic 3 to 1 ratio of military superiority and still have resources to spare. So you cannot take even the slightest military action against them, including raiding parties or trying to expand your territory.
Note that this other larger faction would already have resources to support their population and you would need to gain resources to support any growth. That means land, food, possibly resources like metals, ores, fuel. It's all uphill for you and your allies, but your "enemies" already have sufficient resources to sustain them through anything you can do, retaliate for anything you do and subvert you.
we built a great underground Bunker, and saved all the important people of the country (politicians, military officials, and scientific minds) and hid in our safety bunker for more then 100 years
What a pity they're all dead (and they can't have been that great or clever if they couldn't prevent the war in the first place).
The next generation won't necessarily be good at anything. What exactly do a bunch of politicians and military leaders teach their population of 5000 (and did you include enough young people to produce new generations ?). Personally I'd shoot every politician trying to enter the bunker as they're a waste of space, and you need a bunch of military leaders to lead what army ? With a population of 5000 people there can't be more than 2000 adults capable of waging war or even defense. And these would be the same military leader who helped destroy the world ? These morons provide what useful skills ? You can learn military and political tactics and strategy from a book, and good teachers would be of more practical benefit that a bunch of idiots who caused the problem in the first place.
We want to increase our population in about a timeframe of ~20 years.
Optimistically you have a 4 to 1 ratio of women to men, because men are useless and can't get pregnant. It takes about 18 years to "grow" an adult that's useful at anything, and so to get a bunch of 17 year old or better adults in twenty years would give you only three years (about 36 months) of your initial "crop" of children.
You start with maybe 4000 women and we'll (unrealistically) assume they're all of child bearing age. Let's say they can produce on average 2 children every nine months (unrealistic). Let's also (unrealistically) assume you can support these extra children's development (food, housing, medical, training, etc.).
That's still only about 32,000 poor idiots for the front lines in your insane quest for power.
Your large "enemy", doing nothing, can probably produce a standing army of 60,000 and afford to loose many of them - you can't afford casualties. You still loose (and they have resources).
But more realistic numbers would limit you to a growth of more like 7000, allowing for the need to support the new children (largely unproductive) in a closed environment with limited (probably capped) resources. And you're definitely going to loose then.
numerous larger factions have arisen, most notably the UCR, a government down south with more than 300,000 people!
So even if you start attacking this much larger faction successfully there can be more large factions that will immediately see you as a threat and ally themselves to crush your trouble-making asses for your arrogance. Get the picture ?
You can't win.
In this world you have one option for success : diplomacy and economics.
Forget competing with the larger factions and think allying with them to grow within their economic and political sphere.
If a nuclear war that practically wipes out the entire population doesn't teach you the insanity of competing and viewing neighbors as enemies then you won't survive long.
Likewise your neighbors would be on the look out for the slightest sign of hostility (or arrogant flag-waving self-importance) from you and react very quickly to put you in your place.
It's peace or death. Pick one.