Could Beringia cross the Pacific without being blocked by the USA in
case of conflict or is the ocean large enough for it to not happen?
The Pacific Ocean is not large enough for this to not happen. Outer space is that big. The oceans of Earth are not.
It would be technologically feasible for the modern U.S. to block surface ships in this scenario if it were determined to do so.
It would first use patrol aircraft (like the P-8), or satellites, or a network of small surveillance ships, or some other means to detect surface ship travel. Then, it would deploy aircraft or warships with anti-ship missiles, or nuclear attack submarines to destroy the ships.
For merchant ships, far less advanced technologies (like cutters or frigates that would board merchant ships with small contingents of marines) could do the job.
Realistically, the U.S. could pose a serious barrier to Beringia doing this at any technology level from the 1930s and beyond, given the experiences of World War II, and could pose a meaningful barrier although not nearly as effective a barrier, even with World War I or 1920s technology.
But, it would have to be willing to engage in an act of war in international waters to do so.
Historically, global powers have respected international waters as free for travel except during World Wars and even then only with respect to active combatants in those wars.
Non-nuclear submarines would be hard pressed to travel that far in the Pacific without the support of visible and vulnerable resupply ships with the same vulnerabilities as surface ships, but could otherwise avoid detection.
Nuclear powered submarines could make the trip without being resupplied and could probably avoid detection, at any technology level through the present. But, it is hard to carry a lot of people or freight that way at an affordable cost. It wouldn't be impossible (there have been a handful of nuclear cargo submarines mostly designed for going under the Arctic ice sheet to save time in shipping), but difficult and expensive.
It isn't clear what the relative technology capabilities and relative industrial capabilities of the countries in question would be and what their stomach would be for interdicting Berginian maritime traffic in international waters.
Of course, we do have historical precedents for this kind of thing.
The British in the far northern hemisphere, controlled Australia, New Zealand, South Africa, India, Hong Kong, and the Falkland Islands controlling it all before aircraft were invented. It fought a war in the late 20th century to hold onto the Falkland Islands and won.
The Dutch controlled lots of territory in Indonesia, half a world away and accessible only by sea at the time that this commenced.
Belgium had African colonies.
Russia waged a war with Japan that was a close struggle despite being heavily western tilted in population and economic resources.